# Localizing firearm carriers by identifying human-object pairs

###### Abstract

Visual identification of gunmen in a crowd is a challenging problem, that
requires resolving the association of a person with an object (firearm). We
present a novel approach to address this problem, by defining human-object
interaction (and non-interaction) bounding boxes. In a given image, human and
firearms are separately detected. Each detected human is paired with each
detected firearm, allowing us to create a paired bounding box that contains
both object and the human. A network is trained to classify these paired-
bounding-boxes into human carrying the identified firearm or not. Extensive
experiments were performed to evaluate effectiveness of the algorithm,
including exploiting full pose of the human, hand-keypoints, and their
association with the firearm. The knowledge of spatially localized features is
key to success of our method by using multi-size proposals with adaptive
average pooling. We have also extended a previously firearm detection dataset,
by adding more images and tagging in extended dataset the human-firearm pairs
(including bounding boxes for firearms and gunmen). The experimental results
(78.5 $AP_{hold}$) demonstrate effectiveness of the proposed method.

Index Terms—  Firearms Detection, Gun violence

##  1 Introduction

Gun violence incidents are universal across the globe. These incidents account
for many lives around the world every year [1, 2, 3]. A lot of administrative
steps have been taken to minimize these incidents but despite all the efforts,
the frequency of such events is increasing with time. Due to lack of direction
in efficient technology based systems for firearm control on large scale,
number of casualties have grown over the years. There is an immense need to
come up with a scientific solution to this problem, specially with the huge
number of cameras and other imaging systems available today.

Many countries and private security agencies have deployed surveillance
systems in public places like schools, colleges, parks, and malls for public
safety. These systems require huge workforce to manually look at the footage
from the surveillance feed. The downside of operating manually is that it
requires lots of human efforts with more chances of mistakes, because of lack
of vigilance over long working hours. So, there is a need for a method that
should not only be able to detect firearms but it should also be able to
categorize the human, who is carrying. Such a novel scientific solution can be
embedded in surveillance systems for significant improvement in identifying a
potential gun violence incident.

Fig. 1: Proposed firearm carrier detection network which not only tells the
firearm is carried but also identifies the carrier at image level. Use of
Adaptive Average Pooling allow us to train our model using multi-size
proposals.

In the recent years, use of visual systems for classification [4, 5, 6] and
detection [7, 8, 9, 10, 11, 12, 13, 14, 15] are of great importance. Deep
learning based neural networks [4, 5, 6] have performed quite well in key
computer vision tasks. These systems learn unique visual features associated
with different objects from the given set of images. Many object detection
algorithms are designed to perform relatively well on benchmark datasets like
PASCAL VOC [16] and MS-COCO [17], but to apply these algorithms directly on
objects like firearms is not much effective. Taking base architecture Faster
RCNN [7] for object detection, firearms (gun and rifle) detection has been
improved in OAOD [18].

Identification of firearms is difficult task owing to the wide variety of
firearms available around the globe. Inherent differences in shape and size
make identification of firearms even more challenging. Furthermore, to detect
if a human is holding a firearm, is a tougher task because of the clutter,
small sized guns, and extremely large but thin rifles making their bounding
boxes spread over a larger window which may contain multiple humans. Due to
the overlapping of multiple humans in the firearm box, it is very hard to
determine which human is actually holding the firearm.

Existing work still lacks to identify that who is carrying firearm? So
integrating this module with firearm detection is a novel task addressed in
this work. For that we borrow subset of firearms dataset publicly available
[18] with additional images. New dataset is manually tagged containing humans
and firearms (labels: gun and rifle) bounding boxes with carried and non-
carried labels. Along with the dataset, in this paper, a method is proposed
which detects whether the firearm is being carried by a specific human. Figure
1 gives a framework of final model that classifies human-firearm pair
localization.

We integrate knowledge from different fields to develop a novel solution that
can detect objects like firearms from OAOD and classify associations with
human in images labeling carried. Incorporating OpenPose [19], we use pose
estimation to locate hands and build association of firearm being carried.
Then as baseline results, we use OAOD for firearms detection and Faster RCNN
[7] (trained on MS-COCO [17] with ResNet-101 [5]) for human detection.
Afterwards, enclosed overlap ($IoUs\geq 0.5$) between detected humans and
objects (firearms) is computed and considered true positive if carried (a
metric considered in Human Object Interaction (HOI) systems). To improve from
baseline results, we train classification model from the association annotated
in our dataset of humans and firearms being carried and not-carried. Our model
is based on VGG-16 along with adaptive average pooling (AAP) to handle multi-
size proposals. Moreover, AAP also allow us to capture context with maximum
participation of primary object in an image.

##  2 Related Work

Object Detection: With advancement of deep learning, object detection research
has achieved significant improvement [7, 8, 11, 12, 14, 15]. Generic object
detection may be divided into two main categories: one stage object detectors
(excel in speed) [8, 9, 10, 11, 13, 14, 15] and two stage object detectors
(excel in accuracy) [7, 12, 20]. There is speed-accuracy trade-off [21],
suggesting to gain accuracy and speed at the same time is still a challenging
task.

Firearm Detection: There are much fewer visual systems specifically dedicated
to firearms detection. Javed et al. [18] has developed a weakly supervised
Orientation Aware Object (firearm) Detection system (OAOD), which detects guns
and rifles in an image, based on the orientation information. Instead of
taking oriented Bounding Boxes (BB) during training, axis aligned BB are used.
Olmos et al. [22] has applied Faster RCNN on gun detection problem and their
system detects only handguns. Akcay et al. has adopted different methods
including one and two stage methods for the detection of gun using x-ray
baggage security imagery [23]. None of these methods presents a solution to
identify firearms carriers.

Pose Estimation: Numerous methods have been proposed to estimate human pose in
2D and 3D vision systems [24, 19, 25], but there is no work done that uses
human pose estimation to predict whether a person is firearm carrier. OpenPose
[19] uses part affinities between joints and keypoints to estimate poses.
Simon et al. proposed hands keypoint estimation using multi-view bootstrapping
[24], which estimates keypoints of hands.

Human Object Interaction: The work in Human Object Interaction (HOI) mainly
exploits context and attention mechanism to detect human and object
interaction [26]. This would not help in more complex tasks required for
localizing the actor and interaction recognition. Exploiting different human
based approaches to detect human and object such as pose and action
information may improve HOI performance [27]. By integrating knowledge from
different fields, we propose a novel solution that can detect objects like
firearms (gun and rifle) in images along with being carried by respective
human. This paper introduces first such method which uses human and hand
detection followed by firearm detection to identify firearm carriers in
crowded scenes.

##  3 Baseline Carrier Detection Approaches

In firearm detection research, no work has yet been proposed for the carrier
detection. The OAOD method only detects firearms in the images, while the
current work we integrate firearm location and human information to classify
firearms as being carried or not carried. For firearm classification, consider
three different baseline approaches including two based on identifying the
human hand and the firearm closeness, and the third one is based on overlap of
human bounding boxes and firearm bounding boxes.

###  3.1 Detecting Hands Inside Firearm Bounding-boxes (HiFB)

In this approach firearm bounding box is detected using OAOD algorithm.
Firearm bounding boxes are then input to Multiview Bootstrapping algorithm
[24] to identify the hand keypoints. From the predicted keypoints, we identify
the set of the keypoints having confidence larger than a selected threshold
$\alpha$. If carnality of the set is greater than the two we classify the
bounding box as carried. In our case, $\alpha$ is set to $0.3$. This approach
achieves reasonable accuracy for classifying gun as being carried/not-carried,
however for the case of rifles, accuracy is low. It is because extremely large
bounding boxes of thin and long rifles are not usually hand-centered. These
large boxes cannot be used as the probable hand locations as actual hands will
cover less pixels. In gun bounding boxes, this condition significantly meets
the requirement and hand to box pixels ratio greater as compared to bounding
boxes containing rifles.

###  3.2 Body-pose Conditioned Firearm carried detection (BCFD)

In this approach full human body pose containing body parts is detected using
OpenPose [19] and OAOD [18] is used to independently detect firearms in a
given image. The detected firearms bounding boxes are inspected against the
estimated hand keypoints. The firearm is categorized as carried if a hand
keypoints and firearm location overlap exceeds the threshold $\beta$. The
problem with this method is that, full human body is often not visible in most
cases due to occlusions or partial appearance. If keypoints of elbow and wrist
are not detected then keypoints of hands are also remain undetected.
Therefore, this approach suffers performance degradation correlated with the
performance of the pose detector.

###  3.3 Measuring Overlap of Human and Firearms Bounding-boxes (OHFB)

In this approach, human detection is performed by using Faster-RCNN with
ResNet-101, pre-trained on MS-COCO. The firearm detection is performed using
OAOD algorithm. Intersection of all detected firearms and detected human
bounding boxes is performed. Association between firearm and human is
established by choosing one with maximum overlap. However, associations with
$IoU<0.5$ are removed. This approach’s performance suffers in case of crowded
scene where firearm BB may have larger overlap with a non-carrier (Fig. 2).

##  4 Proposed Human Firearm Pair Detection (HFPD)

In addition to the firearm classification as carried or not carried, we aim to
identify the carrier of the firearm. The methods (Sec. 3.1, 3.2) where we
explicitly try to create association between human pose and firearm, the
pairing fails due to error in the pose estimation. A naive idea, considered in
3.3), is to classify firearm and human as paired on the basses of overlap
between them. Since, this method does not take in consideration the
association of human body with the firearm it fails to achieve good
performance, e.g. in crowded scenes. Therefore, we propose to train a neural-
network which will learn using training data the necessary features to
identify if a particular firearm and human are paired.

Fig. 2: Output results by our proposed model. Firearms are shown in Green
bounding box, humans are shown in Blue bounding box. Pairwise-extended boxes
are ON with respective colors if returned carried by our classification model.

In a given image, human and firearms bounding boxes are separately detected as
discussed in the Section 3.2. Each detected human bounding box is paired with
each detected firearm, allowing us to create a paired bounding box that
contains both the firearm and the human. Some of these paired bounding boxes
contain both firearm and its carrier, in others the pair is not associated. A
network is trained to classify these paired-bounding-boxes into interaction or
non-interaction. For training, the extended boxes are manually labeled which
tell whether the human present in the box is carrying the firearm or not.
Features from these multi-size pairwise-extended boxes are used to learn
associations of humans and firearms. A classification model is trained on
these instances. For the pair classification network, We use VGG-16
(consisting of five VGG blocks of convolutional layers) as feature extraction,
followed by Adaptive Average Pooling (AAP) and two fully connected layers. AAP
allows us to use multi-size proposals without resizing them. The cross entropy
loss function for object classification is defined as:

$\displaystyle
L_{c}(p_{c},g_{c})=\sum_{i=1}^{n_{s}}\sum_{j=1}^{n_{c}}g_{c}(i,j)\text{log}(p_{c}(i,j))$
(1)

where $p_{c}\in\mathcal{R}^{n_{c}}$ is the predicted class probability of
being carried and not-carried and
$g_{c}=\\{\\{1,0\\},\\{0,1\\}\\}\in\mathcal{R}^{n_{c}}$ is the ground truth
class label, $n_{c}=2$ is the number of classes, and $n_{s}$ is the number of
samples in batch.

Table 1: Classification of firearm as carried or not carried Methods | Gun | Rifle | Overall
---|---|---|---
HiFD | 71.9 | 37.5 | 49.2
BCFD | 51.3 | 76.4 | 66.6

##  5 Experiments and Results

###  5.1 Dataset and Annotations

As per our knowledge, we are the first to release dataset that contains
bounding boxes of firearm and humans, with the association between the firearm
and carrier tagged. A subset of images was selected from [18], that contain
multiple humans and one or more firearms. 900 more images were added to the
dataset, bringing the size to 3128. For each image, firearm and human bounding
box are manually annotated, including the pair-bounding box. The pair-bounding
box is labelled 1 if it contains valid interaction, 0 if not. All the
experiments are evaluated on this dataset. Some samples of images from dataset
are shown in Figure 3.

Fig. 3: Samples images from the dataset, Firearms are shown in Orange bounding
box, humans in Pink bounding box, Red bounding box show association of human
and firearm. Negative associations are not shown to avoid clutter.

###  5.2 Comparison of Different Algorithms

We compare the performance of experiments along with the experimental details.
Table 2 shows the experimental results and comparison with the baseline.

####  5.2.1 Measuring Overlap of Human-Firearms Detection

Experiments were conducted using pre-trained models, for humans (pre-trained
Faster RCNN with ResNet-101 and VGG-16 on MS-COCO) and firearms (OAOD)
detection. We share the results on both backbones, which indicate ResNet-101
achieve better detection of humans. The model is evaluated, end-to-end. Both,
the predicted human and firearms bounding boxes must have IoUs $\geq$ 0.5 to
be considered as positive sample for the paired-bounding box. These results
are used as baseline, for human-firearm interaction.

Table 2: Results of Human-Firearms pair identification with and without Adaptive Average Pooling (AAP) ) Methods |  | Backbone
---
HumanDet
$AP_{Ghold}$ | $AP_{Rhold}$ | $AP_{hold}$
Baseline OHFB | VGG-16 | 42.4 | 62.6 | 54.64
| HFPD w/o
---
AAP
VGG-16 | 62.4 | 67.2 | 64.7
| HFPD with
---
AAP
VGG-16 | 64.8 | 73.4 | 69.3
Baseline OHFB | ResNet-101 | 65.0 | 74.5 | 70.6
| HFPD w/o
---
AAP
ResNet-101 | 72.3 | 78.9 | 76.0
| HFPD with
---
AAP
ResNet-101 | 75.3 | 81.1 | 78.5

####  5.2.2 Joint Human-firearm Interaction Bounding Boxes

For this experiment, VGG-16 network is fine-tuned (using ImageNet weights) on
the annotated bounding boxes of humans-firearms association from the proposed
dataset. Note that, we use two fully connected layers of pre-trained VGG
network and add final classification layer over it. That restricts the size of
image that could be input to the network. Adaptive average pooling (AAP) is
used between last convolutional layer and first FC layer to handle multi-size
inputs to the network. Without (w/o) AAP, results are reported with fixed
resized image (224x224) that simple VGG-16 model allows but aspect ratio
disturbs in most of the cases. AAP (output=7x7) is being used to handle multi-
size boxes during training and testing as actual VGG-16 does not allow random
sized input. AAP also helps to capture prominent details with primary object
participation. It is to note that, resizing of multi-size pair-wise extended
boxes are done in a way that long dimension corresponds to 600 and shorter
side adjusts accordingly. It can be seen in Table 2, that using ResNet-101 for
human detection yields better proposal with firearms for our model to
classify. With AAP, learning rate is 0.00001 with batch size = 1 and dropout
(50%) is used while training. We adopt stochastic gradient descent (SGD) to
train the model for 20 epochs, and a momentum of 0.9.

##  6 Conclusion and Future Directions

In this paper we present a novel method for localizing firearm and its carrier
in a crowded scene. The problem is not only challenging but its solutions is
dearly needed in current law-and-order situation where intelligent cameras are
required to perform the surveillance. We exploit human and firearm detection
to create paired-bounding boxes for every possible human-object (firearm) pair
in the image. Reducing our problem to classifying the pair as valid
interaction or not. Employing, adaptive average pooling and multi-size
proposals, we were able to achieve 78.5 $AP_{hold}$ on the test dataset.
Extensive comparative experiments were performed by designing various baseline
strategies, including ones exploring human pose and hand-keypoint information.
In future, we explore the importance of salient areas of firearms and humans
spatially correlated with their respective regions. Existing OAOD dataset [18]
was extended to include more challenging images and were hand-tagged to
identify human-firearm pairs.

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# Nonclassicality of photon-added squeezed vacuum and its decoherence in
thermal environment††thanks: Project supported by the National Natural Science
Foundation of China (Grant Nos 10775097 and 10874174) and the Research
Foundation of the Education Department of Jiangxi Province.

Li-yun Hu1,2 and Hong-yi Fan2
1College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China
Corresponding author. _E-mail address_<EMAIL_ADDRESS>(L-Y Hu).

###### Abstract

We study the nonclassicality of photon-added squeezed vacuum (PASV) and its
decoherence in thermal environment in terms of the sub-Poissonian statistics
and the negativity of Wigner function (WF). By converting the PASV to a
squeezed Hermite polynomial excitation state, we derive a compact expression
for the normalization factor of $m$-PASV, which is an $m$-order Legendre
polynomial of squeezing parameter $r$. We also derive the explicit expression
of WF of $m$-PASV and find the negative region of WF in phase space. We show
that there is an upper bound value of $r$ for this state to exhibit sub-
Poissonian statistics increasing as $m$ increases. Then we derive the explicit
analytical expression of time evolution of WF of $m$-PASV in the thermal
channel and discuss the loss of nonclassicality using the negativity of WF.
The threshold value of decay time is presented for the single PASV.

PACS number(s): 03.67.-a, 03.65. Ud, 42. 50.Dv

##  1 Introduction

Nonclassical states play an important role in quantum computation and quantum
imformation procession [1]. The nonclassicality of quantum states can be well-
described by some nonclassical properties, such as sub-Poissonian photon
ststistics [2], squeezing in one of the quadratures of the field [3], and
negativity of Wigner function (WF) [4]. Especially, the partial negativity of
WF implies the highly nonclassical properties of quantum states and is often
used to describe the decoherence of quantum states. Many experimental schemes
have been proposed to generate nonclassical states of optical field. Among
them, subtracting photons form and/or adding photons to quantum states have
been paid much attention because these fields exhibit an abundant of
nonclassical properties and may give access to a complete engineering of
quantum states and to fundamental quantum phenomena [5, 6, 7, 8, 10, 11, 12,
13]. For example, a single photon-subtraction squeezed vacuum (PSSV) has been
experimentally prepared with a pulsed and continuous wave squeezed vacuum [11,
12, 13]. It is very similar to quantum superpositions of coherent states with
small amplitudes [14, 15]. As another example, a single-photon addition was
experimentally performed by Zavatta et al [6], which unveils the nonclassical
features associated with the excitation of a classical coherent field by a
single light quantum. For the single PSSV, its nonclassical properties and
decoherence was investigated theoretically in two different decoherent
channels (amplitude decay and phase damping) by Biswas and Agarwal [16]. They
indicated that the WF losses its non-Gaussian nature and becomes Gaussian at
long times in amplitude decay case.

On the other hand, the combination of the photon addition and subtraction has
been successfully demonstrated in Ref.[8, 9]. In Ref.[8], photon addition and
subtraction experimentally have been employed to probe quantum commutation
rules by Parigi et al. In fact, they have implemented simple alternated
sequences of photon creation (addition) and annihilation (subtraction) on a
thermal field and observed the noncommutativity of the creation and
annihilation operators. It is interesting to notice that subtracting or adding
one photon from/to pure squeezed vacuums can generate the same output state,
i.e., squeezed single-photon state [17]. However, the resulting states
obtained by successive photon subtractions or additions are different from
each other. For instance, successive two-photon additions [$a^{{\dagger}2}$]
and successive two-photon subtractions [$a^{2}$] will result in the same state
produced by using subtraction-addition ($a^{{\dagger}}a$) and addition-
subtraction ($aa^{{\dagger}}$), (also see section 2 below) respectively. As
far as we know, the nonclassicality and decoherence of arbitrary number
photon-added squeezed vacuum states (PASV) in a dissipative channel has not
been derived analytically in the literature before.

In this paper, we shall investigate the nonclassical properties and
decoherence of single-mode PASV which is optically produced single-mode non-
Gaussian states. This work is arranged as follows. In section 2, we introduce
the single-mode PASV and discuss its nonclassicality in terms of sub-
Poissonian statistics and the negativity of its Wigner function (WF). By
converting the PASV to a squeezed Hermite polynomial excitation state, we
derive a compact expression for the normalization factor of PASV, which is an
$m$-order Legendre polynomial of the squeezing parameter $r$, where $m$ is the
number of added photons; and then derive the explicit analytical expression of
WF for any photon-added number $m$ and find the negative region of WF in phase
space. We also show that there is an upper bound value of $r$ for this state
to exhibit sub-Poissonian statistics which increases as $m$ increases. In
section 3, we derive the explicit analytical expression of time evolution of
WF of the arbitrary PASV in the thermal channel and discuss the loss of
nonclassicality in reference of the negativity of WF. The threshold value of
the decay time corresponding to the transition of the WF from partial negative
to completely positive definite is obtained at the center of the phase space,
which is independent of the squeezing parameter. We show that the WF for
single PASV has always negative value for all parameters $r$ if the decay time
$\kappa t<\frac{1}{2}\ln\frac{2\bar{n}+2}{2\bar{n}+1}\ $(see Eq.(40) below),
where $\bar{n}$ denotes the average thermal photon number in the environment
with dissipative coefficient $\kappa$. Conclusions are involved in the last
section.

##  2 Single-mode Photon added squeezed vacuum state

Various photon states have been generated by the micromaser and WFs of some
cavity fields can be measured by a scheme based on interaction between cavity
fields and atoms [18, 19]. As described in Ref. [20] when an excited atom
passes through a cavity field which is in a squeezed vacuum state then their
interaction may produce photon addition (excitation) on the squeezed vacuum
state—the excited squeezed vacuum state.

Therectically, the single-mode PASV can be obtained by repeatedly operating
the photon creation operator $a^{{\dagger}}$ on a squeezed vacuum state
$S\left(r\right)\left|0\right\rangle$, i.e.,

$\left|r,m\right\rangle\equiv
N_{r,m}a^{{\dagger}m}S\left(r\right)\left|0\right\rangle,$ (1)

where $\left|0\right\rangle$ is single mode vacuum, and $N_{r,m}\ $is the
normalization constant to be determined, $a^{{\dagger}}$ is the Bose creation
operator, and $S\left(r\right)$ is the single-mode squeezing operator
$S\left(\lambda\right)=\exp[\frac{1}{2}\left(ra^{\dagger 2}-ra^{2}\right)]$
[21, 22] with $r$ being the squeezing parameter.

###  2.1 Single-mode PASV as the squeezed Hermite polynomial excitation state

Under the transform of $S\left(\lambda\right)$ we see
$S^{{\dagger}}\left(r\right)a^{\dagger}S\left(r\right)=a^{\dagger}\cosh
r+a\sinh r,$ thus we can reform Eq.(1) as

$\displaystyle\left|r,m\right\rangle$
$\displaystyle=N_{r,m}S\left(r\right)S^{{\dagger}}\left(r\right)a^{{\dagger}m}S\left(r\right)\left|0\right\rangle$
$\displaystyle=N_{r,m}S\left(r\right)\left(a^{{\dagger}}\cosh r+a\sinh
r\right)^{m}\left|0\right\rangle.$ (2)

On the other hand, using the operator identity [23]

$\left(a\mu+\nu
a^{\dagger}\right)^{m}=\left(-i\sqrt{\frac{\mu\nu}{2}}\right)^{m}\colon
H_{m}\left(i\sqrt{\frac{\mu}{2\nu}}a+i\sqrt{\frac{\nu}{2\mu}}a^{{\dagger}}\right)\colon,$
(3)

where $H_{m}\left(x\right)$ is $m$-order single variable Hermite polynomial
whose definition is

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

we have

$\displaystyle\left|r,m\right\rangle$
$\displaystyle=\frac{\left(-i\right)^{m}}{2^{m}}N_{r,m}\sinh^{m/2}2rS\left(r\right)\colon
H_{m}\left(i\sqrt{\frac{\tanh r}{2}}a+i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\colon\left|0\right\rangle$
$\displaystyle=\frac{\left(-i\right)^{m}}{2^{m}}N_{r,m}\sinh^{m/2}2rS\left(r\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle,$ (4)

which indicates that the PASV is equivalent to a squeezed Hermite-excited
vacuum state.

Further using the generating function of $H_{m}\left(x\right)$,

$H_{m}\left(x\right)=\left.\frac{\partial^{m}}{\partial
t^{m}}\exp\left(2xt-t^{2}\right)\right|_{t=0},$ (5)

and Eq.(4), the normalization factor $N_{r,m}$ can be derived by

$\displaystyle 1$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\left\langle
0\right|H_{m}\left(-i\sqrt{\frac{\coth
r}{2}}a\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\frac{\partial^{2m}}{\partial\tau^{m}\partial
t^{m}}e^{-t^{2}-\tau^{2}}\left.\left\langle 0\right|e^{-i\sqrt{2\coth
r}a\tau}e^{i\sqrt{2\coth
r}a^{{\dagger}}t}\left|0\right\rangle\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\frac{\partial^{2m}}{\partial\tau^{m}\partial
t^{m}}\left.\exp\left(-t^{2}-\tau^{2}+2\tau t\coth
r\right)\right|_{\tau=t=0},$ (6)

where in the last step we have used the Baker-Hausdorff formula $e^{\mu
a}e^{\nu a^{\dagger}}=e^{\nu a^{\dagger}}e^{\mu a}e^{\mu\nu}$. Using the newly
found generating function of Legendre polynomial [24, 25] (see Appendix A),

$\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+\frac{2x\tau
t}{\sqrt{x^{2}-1}}\right)\right|_{t,\tau=0}=\frac{2^{m}m!}{\left(x^{2}-1\right)^{m/2}}P_{m}\left(x\right),$
(7)

we can derive the compact expression for $N_{r,m}$, i.e.,

$N_{r,m}^{-2}=m!\cosh^{m}rP_{m}\left(\cosh r\right),$ (8)

which is just the result in Ref.[20] derived by the mathematical induction
method. In particular, when $m=1,$ ($H_{1}\left(x\right)=2x$) i.e., the
single-photon added case, we see that
$\left|r,1\right\rangle=S\left(r\right)a^{{\dagger}}\left|0\right\rangle$ is
just a squeezed single-photon state; on the other hand, for the single-photon
subtracted case [17, 26], the state is
$aS\left(r\right)\left|0\right\rangle=S\left(r\right)S^{\dagger}\left(r\right)aS\left(r\right)\left|0\right\rangle=S\left(r\right)\left(a\cosh
r+a^{{\dagger}}\sinh r\right)\left|0\right\rangle\rightarrow
S\left(r\right)a^{{\dagger}}\left|0\right\rangle,$ which indicates that adding
a single-photon to the squeezed state has the same impact as annihilating a
photon from the squeezed state. While for $m\geqslant 2,$ the case is not true
(also see Fig.1). For example, successive two-photon additions
[$a^{{\dagger}2}$] and successive two-photon subtractions [$a^{2}$] will
result in the same state produced by using subtraction-addition
($a^{{\dagger}}a$) and addition-subtraction ($aa^{{\dagger}}$), respectively,
i.e., $a^{{\dagger}2}S\left(r\right)\left|0\right\rangle\rightarrow
a^{{\dagger}}aS\left(r\right)\left|0\right\rangle,a^{2}S\left(r\right)\left|0\right\rangle\rightarrow
aa^{{\dagger}}S\left(r\right)\left|0\right\rangle.$ In Ref.[27], two PSSV is
used to generate the squeezed superposition of coherent states with high
fidelities and large amplitudes.

Combining Eqs.(1) and (8) we can conveniently calculate the average photon
number $a^{{\dagger}}a$ in PASV,

$\displaystyle\left\langle r,m\right|a^{{\dagger}}a\left|r,m\right\rangle$
$\displaystyle=\left\langle r,m\right|aa^{{\dagger}}\left|r,m\right\rangle-1$
$\displaystyle=\frac{N_{r,m}^{2}}{N_{r,m+1}^{2}}-1$
$\displaystyle=\left(m+1\right)\zeta\frac{P_{m+1}\left(\zeta\right)}{P_{m}\left(\zeta\right)}-1,$
(9)

where we denote $\zeta=\cosh r$ for simplicity, and

$\displaystyle\left\langle
r,m\right|a^{{\dagger}2}a^{2}\left|r,m\right\rangle$
$\displaystyle=\left\langle
r,m\right|\left(a^{2}a^{{\dagger}2}-4aa^{{\dagger}}+2\right)\left|r,m\right\rangle$
$\displaystyle=\frac{N_{r,m}^{2}}{N_{r,m+2}^{2}}-4\frac{N_{r,m}^{2}}{N_{r,m+1}^{2}}+2$
$\displaystyle=\left(m+1\right)\zeta\left\\{\left(m+2\right)\zeta\frac{P_{m+2}\left(\zeta\right)}{P_{m}\left(\zeta\right)}-4\frac{P_{m+1}\left(\zeta\right)}{P_{m}\left(\zeta\right)}\right\\}+2,$
(10)

thus the Mandel’s $\mathcal{Q}$-parameter can be obtained by substituting
Eqs.(9) and (10) into $\mathcal{Q}\equiv\frac{\left\langle a^{\dagger
2}a^{2}\right\rangle}{\left\langle a^{{\dagger}}a\right\rangle}-\left\langle
a^{{\dagger}}a\right\rangle$. In particular, for single-photon-added case
$m=1,$ Eqs.(9) and (10) reduce to

$\displaystyle\left\langle r,1\right|a^{{\dagger}}a\left|r,1\right\rangle$
$\displaystyle=3\cosh^{2}r-2,$ (11) $\displaystyle\left\langle
r,1\right|a^{{\dagger}2}a^{2}\left|r,1\right\rangle$
$\displaystyle=\frac{3\left(3+2\tanh^{2}r\right)}{\left(\coth r-\tanh
r\right)^{2}},$ (12)

thus the $\mathcal{Q}$-parameter with $m=1$ is given by

$\mathcal{Q=}\frac{3\sinh^{2}2r}{3\cosh 2r-1}-1.$ (13)

From Eq.(13) we find that $\mathcal{Q}$ becomes negative for $m=1$ which is
satisfied for the squeezing parameter $r\lesssim 0.46$ similar to the result
of $\mathcal{Q}$ for single-photon subtracted case [16]. In order to see
clearly the variation of $\mathcal{Q}$-parameter with $r$, we show the plots
of $\mathcal{Q}$-parameter in Fig.1, from which one can clearly see that
$\mathcal{Q}$-parameter becomes negetive ($m\neq 0)$ when $r$ is less than a
certain threshold value which increases as $m$ increases; while for $m=0,$
$\mathcal{Q}$ is always positive. This implies that the nonclassicality is
enhanced by adding photon to squeezed state. We should emphasize that the WF
has negative region for all $r,$ and thus the PASV is nonclassical. In our
following work, we pay attention to the (ideal) PASV in a thermal channel.

Figure 1: (Color online) The $Q$-parameter as the function of squeezing
parameter $r$ for different $m=0,1,2,3,4,29,30.$

###  2.2 Wigner function of PASV

In order to discuss the decoherence properties of PASV in thermal environment,
in this subsection we shall derive the analytical expression of Wigner
function for PASVS. For single-mode case, the Wigner operator is defined as
[28]

$\Delta\left(q,p\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}du\left|q-\frac{u}{2}\right\rangle\left\langle
q+\frac{u}{2}\right|e^{-ipu},$ (14)

where $\left|q\right\rangle$ is the coordinate representation,
$Q\left|q\right\rangle=q\left|q\right\rangle$. Thus the Wigner function of
PASVS $\left|r,m\right\rangle$ can be calculated by
$W\left(q,p\right)=\left\langle
r,m\right|\Delta\left(q,p\right)\left|r,m\right\rangle.$ Using Eq.(4), we can
see

$W\left(q,p\right)=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\left\langle
0\right|H_{m}\left(-i\sqrt{\frac{\coth
r}{2}}a\right)S^{{\dagger}}\left(r\right)\Delta\left(q,p\right)S\left(r\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle.$ (15)

On the other hand, noticing that the single-mode squeezing operator
$S^{{\dagger}}\left(r\right)$ has its natural expression in coordinate
representation [29], i.e.,
$S^{{\dagger}}\left(r\right)=e^{r/2}\int_{-\infty}^{\infty}dq\left|qe^{r}\right\rangle\left\langle
q\right|,$ leading to
$S^{{\dagger}}\left(r\right)\left|q\right\rangle=e^{-r/2}\left|qe^{-r}\right\rangle,$
it then follows that

$\displaystyle
S^{{\dagger}}\left(r\right)\Delta\left(q,p\right)S\left(r\right)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\left(ue^{-r}\right)\left|qe^{-r}-\frac{ue^{-r}}{2}\right\rangle\left\langle
qe^{-r}+\frac{ue^{-r}}{2}\right|e^{-ipu}$
$\displaystyle=\Delta\left(qe^{-r},pe^{r}\right).$ (16)

Then substituting Eq.(16) into Eq.(15) and using Eq.(5) as well as the
coherent state representation of Wigner operator,

$\Delta\left(q,p\right)\rightarrow\Delta\left(\alpha,\alpha^{\ast}\right)=\frac{e^{2\left|\alpha\right|^{2}}}{\pi}\int\frac{d^{2}z}{\pi}\left|z\right\rangle\left\langle-z\right|e^{-2\left(z\alpha^{\ast}-z^{\ast}\alpha\right)},$
(17)

where $\alpha=(q+ip)/\sqrt{2}$ and
$\left|z\right\rangle=\exp(za^{\dagger}-z^{\ast}a)\left|0\right\rangle$ is the
coherent state [30, 31], we can put Eq.(15) into the following form,

$\displaystyle W\left(q,p\right)$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\left\langle
0\right|H_{m}\left(-i\sqrt{\frac{\coth
r}{2}}a\right)\Delta\left(qe^{-r},pe^{r}\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}-\tau^{2}}\left.\left\langle
0\right|e^{-i\sqrt{2\coth
r}at}\Delta\left(qe^{-r},pe^{r}\right)e^{i\sqrt{2\coth
r}a^{{\dagger}}\tau}\left|0\right\rangle\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{r,m}^{2}e^{2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}-\tau^{2}}\left.\int\frac{d^{2}z}{\pi}e^{-\left|z\right|^{2}-\left(i\sqrt{2\coth
r}t+2\beta^{\ast}\right)z+\left(2\beta-i\sqrt{2\coth
r}\tau\right)z^{\ast}}\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{r,m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\left.\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}+2\bar{\beta}t-\tau^{2}+2\allowbreak\bar{\beta}^{\ast}\tau-2t\tau\coth
r}\right|_{\tau=t=0},$ (18)

where

$\bar{\beta}=-i\beta\sqrt{2\coth
r},\beta=(qe^{-r}+ipe^{r})/\sqrt{2}=\alpha\cosh
r-\allowbreak\alpha^{\ast}\sinh r,$ (19)

and in the last step, we have used the integration formula

$\int\frac{d^{2}z}{\pi}e^{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(20)

In order to further simplify Eq.(18), expanding the exponential item
$e^{-2t\tau\coth r}$ as sum series and using Eq.(5) we have

$\displaystyle W\left(q,p\right)$
$\displaystyle=\frac{N_{m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\sum_{l=0}^{\infty}\frac{\left(-2\coth
r\right)^{l}}{2^{2l}l!}\frac{\partial^{2l}}{\partial\left(\bar{\beta}\right)^{l}\partial\left(\bar{\beta}^{\ast}\right)^{l}}\left.\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}+2\bar{\beta}t-\tau^{2}+2\bar{\beta}^{\ast}\tau}\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\sum_{l=0}^{\infty}\frac{\left(-2\coth
r\right)^{l}}{2^{2l}l!}\frac{\partial^{2l}}{\partial\left(\bar{\beta}\right)^{l}\partial\left(\bar{\beta}^{\ast}\right)^{l}}H_{m}(\bar{\beta})H_{m}(\bar{\beta}^{\ast}).$
(21)

Noticing the recurrence relations of $H_{m}(x)$,

$\frac{\mathtt{d}}{\mathtt{d}x^{l}}H_{m}(x)=\frac{2^{l}m!}{\left(m-l\right)!}H_{m-l}(x),$
(22)

then the Wigner function of $\left|r,m\right\rangle$ is given by

$\displaystyle W\left(q,p\right)$
$\displaystyle=\frac{N_{m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\sum_{l=0}^{m}\frac{\left(m!\right)^{2}\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}(\bar{\beta})\right|^{2}$
$\displaystyle=\frac{1}{\pi}\frac{e^{-2\left|\beta\right|^{2}}\sinh^{m}r}{2^{m}P_{m}\left(\cosh
r\right)}\sum_{l=0}^{m}\frac{m!\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}(\bar{\beta})\right|^{2},$
(23)

where $\beta$ and $\bar{\beta}$ are shown in Eq.(19). Eq.(23) seems a new
result (not reported in the literature before), related to single-variable
Hermite polynomials. In particular, when the photon-added number $m=0,1$ and
noticing that $P_{0}\left(\cosh r\right)=1$ and $P_{1}\left(\cosh
r\right)=\cosh r,$ Eq.(23) reduce to, respectively,

$\displaystyle W_{m=0}\left(q,p\right)$
$\displaystyle=\frac{1}{\pi}e^{-(q^{2}e^{-2r}+p^{2}e^{2r})},$ (24)
$\displaystyle W_{m=1}\left(q,p\right)$
$\displaystyle=\frac{1}{\pi}\left\\{2(q^{2}e^{-2r}+p^{2}e^{2r})-1\right\\}e^{-(q^{2}e^{-2r}+p^{2}e^{2r})}.$
(25)

Eq.(24) is just the WF of squeezed vacuum state, a Gaussian in phase space;
while Eq.(25) corresponds to a non-Gaussian WF in phase space due to the
presence of non-Gaussian item $2(q^{2}e^{-2r}+p^{2}e^{2r})-1$. It is clear
that the function $W_{m=1}\left(q,p\right)$ becomes negative in phase space,
when $2(q^{2}e^{-2r}+p^{2}e^{2r})<1$.

Using Eq.(23) we show the plots of WF in the phase space in Figs.2 for
different squeezing parameters $r$ and photon-added numbers $m$. One can see
clearly that there is some negative region of the WF in the phase space which
implies the nonclassicality of this state. In addition, the squeezing effect
in one of the quadratures is clear in the plots (see Figs.2a and Figs.2b),
which is another evidence of the nonclassicality of this state. The WF has its
minimum value for $m=1,3$ at the center of phase space $\left(q=p=0\right)$
(see Fig.2(a) and (d)). The case is not true for $m=2$ (see Fig2.c).

Figure 2: (Color online) The Wigner functions of PASV for (a) $m=1,r=0.3,$(b)
$m=1,r=0.8,$(c) $m=2,r=0.3$ and (d) $m=3,r=0.3.$

##  3 Decoherence of PASV in thermal environment

###  3.1 Model of Decoherence

In this section, we consider how this single-mode state (1) evolves at the
presence of thermal environment. In thermal channel, the evolution of the
density matrix for the $m$-PASV can be described by [32]

$\frac{d\rho}{dt}=\kappa\left(\bar{n}+1\right)\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho
a^{\dagger}a\right)+\kappa\bar{n}\left(2a^{\dagger}\rho
a-aa^{\dagger}\rho-\rho aa^{\dagger}\right),$ (26)

where $\kappa$ represents the dissipative coefficient and $\bar{n}$ denotes
the average thermal photon number of the environment. When $\bar{n}=0,$
Eq.(26) reduces to the master equation describing the photon-loss channel. The
corresponding time evolution density operator can be obtained as

$\rho_{r,m}\left(t\right)=e^{\kappa
t+\Gamma_{0}}\sum_{k,l=0}^{\infty}M_{k,l}\rho_{0}M_{k,l}^{\dagger},$ (27)

where $\rho_{0}=\left|r,m\right\rangle\left\langle r,m\right|$ is the initial
density matrix; $M_{k,l}$ and $M_{k,l}^{{\dagger}}$ are Hermite conjugated
operators (Kraus operator) with each other,

$M_{k,l}=e^{\left(\kappa
t+\Gamma_{0}\right)/2}\sqrt{\frac{\Gamma_{-}^{k}\Gamma_{+}^{l}e^{-2l\Gamma_{0}}}{k!l!}}e^{\Gamma_{0}a^{\dagger}a}a^{\dagger
l}a^{k},$ (28)

and $\Gamma_{+},\Gamma_{-}$ and $\Gamma_{0}$ are determined by

$T=1-e^{-2\kappa t},\text{ }\Gamma_{+}=\frac{\bar{n}T}{\bar{n}T+1},\text{
}\Gamma_{-}=\frac{\left(\bar{n}+1\right)T}{\bar{n}T+1},\text{
}\Gamma_{0}=\ln\frac{e^{-\kappa t}}{\bar{n}T+1}.$ (29)

It is not difficult to prove the $M_{k,l}$ obeys the normalization condition
$\sum_{k,l=0}^{\infty}M_{k,l}^{{\dagger}}M_{k,l}=1$ by using the technique of
integration within an ordered products of operators [33, 34].

###  3.2 Wigner function of the PASV in a thermal channel

The evolution formula of WF of the PASV can be derived as follows [35]

$W\left(\zeta,\zeta^{\ast},t\right)=\frac{2}{\left(2\bar{n}+1\right)T}\int\frac{d^{2}\alpha}{\pi}W\left(\alpha,\alpha^{\ast},0\right)e^{-2\frac{\allowbreak\left|\zeta-\alpha
e^{-\kappa t}\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}},$ (30)

where $W\left(\alpha,\alpha^{\ast},0\right)$ is the WF of the initial state.
Eq.(30) is just the evolution formula of WF in thermal channel. Thus the WF at
any time can be obtained by performing the integration when the initial WF is
known.

In a similar way to deriving Eq.(23), substituting Eq.(23) into Eq.(30) and
using the generating function of single-variable Hermite polynomials (5) and
Eq.(20), we finally obtain (see appendix B)

$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{2\sinh^{m}r}{\pi\sqrt{C}\left(2\bar{n}+1\right)T}\frac{m!e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}}{2^{m}P_{m}\left(\cosh
r\right)}e^{\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}$
$\displaystyle\times\sum_{l=0}^{m}\sum_{k=0}^{m-l}\frac{\left(-2\coth
r\right)^{l}G^{m-l-k}F^{k}}{l!k!\left[\left(m-l-k\right)!\right]^{2}}\left|H_{m-l-k}(E/\sqrt{G})\right|^{2}.$
(31)

where we have set

$\displaystyle A$ $\displaystyle=\frac{2e^{-2\kappa
t}}{\left(2\allowbreak\bar{n}+1\right)T}+2\cosh 2r,\text{ }$ (32)
$\displaystyle B$ $\displaystyle=\frac{2e^{-\kappa
t}\zeta}{\left(2\allowbreak\bar{n}+1\right)T},\text{
}C=A^{2}-4\sinh^{2}2r,\text{ }$ (33) $\displaystyle D$
$\displaystyle=\sqrt{2\coth r}i\left(B^{\ast}\sinh r-B\cosh r\right),$ (34)
$\displaystyle E$ $\displaystyle=\frac{1}{C}\left(AD-2D^{\ast}\sinh
2r\right),$ (35) $\displaystyle F$ $\displaystyle=\frac{8}{C}\left(A\cosh
2r\coth r-4\cosh^{2}r\sinh 2r\right),$ (36) $\displaystyle G$
$\displaystyle=1-\frac{16}{C}\frac{e^{-2\kappa
t}}{\left(2\allowbreak\bar{n}+1\right)T}\cosh^{2}r.$ (37)

Eq.(31) is just the analytical expression of WF for the PASV in thermal
channel. It is obvious that the WF loss its Gaussian property due to the
presence of single-variable Hermite polynomials. In particular, at the initial
time ($t=0$), noting $\sqrt{C}\left(2\bar{n}+1\right)T\rightarrow 2,$
$G\rightarrow 1,$ $F/G\rightarrow 0,$ and
$E/\sqrt{G}\rightarrow\bar{\beta}=-i\sqrt{2\coth r}\left(\zeta\cosh
r-\zeta^{\ast}\sinh r\right),$ as well as
$\frac{A}{C}\left|B\right|^{2}-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}\rightarrow-2\left|\zeta\right|^{2}\cosh
2r,\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)\rightarrow\left(\zeta^{2}+\zeta^{\ast
2}\right)\sinh 2r$, Eq.(31) just does reduce to Eq.(23), i.e., the WF of the
PASV. On the other hand, when $\kappa t\rightarrow\infty,$ noticing that
$T\rightarrow 1,B\rightarrow 0,C\rightarrow 4,D\rightarrow 0,$
$E/\sqrt{G}\rightarrow 0,G\rightarrow 1,$ $F\rightarrow\allowbreak 4\coth
r,$as well as
$H_{m}\left(0\right)=\left(-1\right)^{j}\frac{m!}{j!}\delta_{m,2j},$ then
Eq.(31) becomes
$\allowbreak\frac{1}{\pi\left(2\bar{n}+1\right)}e^{-\frac{2\left|\zeta\right|^{2}}{2\allowbreak\bar{n}+1}}$,
which is independent of photon-addition number $m$ and corresponds to the WF
of thermal state with mean thermal photon number $\bar{n}$. This indicates
that the system state reduces to a thermal state after an enough long time
interaction with the environment.

In addition, for the case of $m=0,$ single-mode squeezed vacuum, Eq.(31) just
becomes ($H_{0}(x)=1$)

$W_{m=0}\left(\zeta,\zeta^{\ast},t\right)=\mathfrak{N}e^{-\mathfrak{D}\left|\zeta\right|^{2}+\mathfrak{E(}\zeta^{\ast}{}^{2}+\zeta{}^{2})},$
(38)

where $\mathfrak{N}=\frac{2}{\pi\sqrt{C}\left(2\bar{n}+1\right)T}$ is the
normalization factor,
$\mathfrak{D}=\frac{2}{\left(2\allowbreak\bar{n}+1\right)T}-\frac{A\mathfrak{E}}{\sinh
2r},\mathfrak{E}\mathfrak{=}\frac{4e^{-2\kappa t}\sinh
2r}{\left[\left(2\allowbreak\bar{n}+1\right)T\right]^{2}C},$ Eq.(38) denotes a
Gaussian distribution function— the WF of single-mode squeezed vacuum in the
thermal channel; while for $m=1$, single–photon added case, its WF in the
thermal channel is given by ($H_{1}(x)=2x$)

$W_{m=1}\left(\zeta,\zeta^{\ast},t\right)=\frac{4\left|E\right|^{2}+F-2\coth
r}{\pi\sqrt{C}\left(2\bar{n}+1\right)T\coth
r}e^{\frac{A}{C}\left|B\right|^{2}-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}.$ (39)

In Fig.3, the WFs of the PASV with $m=1$ are depicted in phase space with
$r=0.3$ and $\bar{n}=1$ for several different $\kappa t.$ It is easy to see
that the negative region of WF gradually diminishes as the time $\kappa t$
increases. Actually, from Eq.(33) one can see that $C>0$, so at the center of
the phase space ($\alpha=\alpha^{\ast}=0$), when $F<2\coth r$ leading to the
following condition:

$\kappa t<\kappa t_{c}\equiv\frac{1}{2}\ln\frac{2\bar{n}+2}{2\bar{n}+1},$ (40)

which is independent of the squeezing parameter $r$, there always exist
negative region for WF in phase space and the WF of PASV is always positive in
the whole phase space when $\kappa t\ $exceeds the threshold value $\kappa
t_{c}$.

In Figs. 4 and 5, we plot the variation of WF in phase space for different
$\bar{n}$ and $r,$ respectively. It is found that the partial negativity of WF
decreases gradually as $\bar{n}$ (or $r$) increases for a given time. The
squeezing effect in one of the quadrature is shown in Fig.5. In addition, for
the case of large squeezing value $r$, the single-photon added squeezed state
becomes similar to a Schodinger cat state (see Fig.6). The WF becomes Gaussian
with the time evolution. In principle, by using the explicit expression of WF
in Eq.(31), we can draw the WF distribution for any photon-added case in phase
space. For instance, for $m=2,$ there are two negative regions of the WF,
which differs from the case of single PASV (see Fig.7). The absolute value of
the negative minimum of the WF decreases as $\kappa t$ increases, which leads
to the complete absence of partial negative region.

##  4 Conclusions

The nonclassical properties and decoherence of single-mode PASV in a thermal
environment have been investigated. A compact expression for the normalization
factor of PASV is derived by converting the PASV to a squeezed Hermite
polynomial excitation state. It is shown that the normalization factor is just
an $m$-order Legendre polynomial of the squeezing parameter $r$. We also
derived the explicit analytical expression of WF for any photon-added number
$m$ and found the negative region of WF in phase space. We also show that
there is an upper bound value of $r$ for this state to exhibit sub-Poissonian
statistics which increases as $m$ increases. Then we considered the effects of
decoherence to the nonclassicality of PASV when interacting with thermal
environment. For arbitrary number PASV, we derived the explicit analytical
expression of time evolution of WF and presented the loss of nonclassicality
in reference of the negativity of WF. The threshold value of the decay time
corresponding to the transition of the WF from partial negative to completely
positive definite is obtained. We find that the WF has always negative value
for all parameters $r$ if the decay time $\kappa
t<\frac{1}{2}\ln\frac{2\bar{n}+2}{2\bar{n}+1}$ for single PASV.

ACKNOWLEDGEMENTS: Work supported by the National Natural Science Foundation of
China (Grant Nos 10775097 and 10874174) and the Research Foundation of the
Education Department of Jiangxi Province.

Figure 3: (Color online) The Wigner functions of single-photon-added squeezed
vacuum states in phase space for $r=0.3,\bar{n}=1$ at $(a)$ $\kappa
t=0.05,(b)$ $\kappa t=0.1,(c)$ $\kappa t=0.2\ $and $(d)$ $\kappa t=0.5.$
Figure 4: (Color online) The Wigner functions of single-photon-added squeezed
vacuum states in phase space for $r=0.3$ and $\kappa t=0.05$ with $(a)$
$\bar{n}=0,(b)$ $\bar{n}=1,(c)$ $\bar{n}=2,(d)$ $\bar{n}=5.$ Figure 5: (Color
online) The Wigner functions of single-photon-subtracted squeezed vacuum
states in phase space for $\bar{n}=0.1$ and $\kappa t=0.05$ with $(a)$
$r=0.03,(b)$ $r=0.5,(c)$ $r=0.8,(d)$ $r=1.5.??$ Figure 6: (Color online)
Wigner function of single photon-added squeezed vacuum states in phase space
for $r=0.8,\bar{n}=0$: $(a)$ $\kappa t=0.1,(b)$ $\kappa t=0.2,(c)$ $\kappa
t=0.3,\ $and $(d)$ $\kappa t=0.7.$ Figure 7: (Color online) The Wigner
functions of photon-added squeezed vacuum states in phase space for
$r=0.7,\bar{n}=1,m=2$: $(a)$ $\kappa t=0.1,(b)$ $\kappa t=0.2.$

Appendix A: Derivation of Eq.(7)

Recalling that the newly found expression of Legendre polynomial [24, 25]

$x^{m}\sum_{{l}=0}^{\left[m/2\right]}\frac{m!}{2^{2{l}}\left({l}!\right)^{2}\left(m-2{l}\right)!}\left(1-\frac{1}{x^{2}}\right)^{{l}}=P_{m}\left(x\right),$
(A1)

which is equivalent to the well-known Legendre polynomial’s expression [36]

$P_{m}\left(x\right)=\sum_{l=0}^{[m/2]}\left(-1\right)^{l}\frac{\left(2m-2l\right)!}{2^{m}l!\left(m-l\right)!\left(m-2l\right)!}x^{m-2l},$
(A2)

though it is different in form from Eq.(A1), they actually are equal. For
instance, what we list in the following equations (the left) from Eq.(A1) are
equal to what we list from (A2) (the right)

$\displaystyle m$ $\displaystyle=0,\text{ }P_{0}\left(x\right)=1;\text{ \ }$
$\displaystyle\text{\ }m$ $\displaystyle=1,\text{ }P_{1}\left(x\right)=x;$
$\displaystyle m$ $\displaystyle=2,\text{
}P_{2}\left(x\right)=x^{2}\left[1+\frac{1}{2}\left(1-\frac{1}{x^{2}}\right)\right]=\frac{3}{2}x^{2}-\frac{1}{2};$
$\displaystyle m$ $\displaystyle=3,\text{
}P_{3}\left(x\right)=x^{3}\left[1+\frac{3}{2}\left(1-\frac{1}{x^{2}}\right)\right]=\frac{5}{2}x^{3}-\frac{3}{2}x;$
(A3)

and

$\displaystyle m$ $\displaystyle=4,\text{
}P_{4}\left(x\right)=x^{4}\left[1+3\left(1-\frac{1}{x^{2}}\right)+\frac{3}{8}\left(1-\frac{1}{x^{2}}\right)^{{2}}\right]=\frac{1}{8}\left(35x^{4}-30x^{2}+3\right);$
$\displaystyle m$ $\displaystyle=5,\text{
}P_{5}\left(x\right)=x^{5}\left[1+5\left(1-\frac{1}{x^{2}}\right)+\allowbreak\frac{15}{8}\left(1-\frac{1}{x^{2}}\right)^{{2}}\right]=\frac{1}{8}\left(63x^{5}-70x^{3}+15x\right);$
(A4)

We emphasize that the new form in Eq.(A1) cannot be directly obtained by some
series summation rearrangement technique from the orginal definition (A2).

On the other hand, noting the following relation

$\displaystyle\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+2x\tau
t\right)\right|_{t,\tau=0}$
$\displaystyle=\sum_{n,l,k=0}^{\infty}\frac{\left(-\right)^{n+l}}{n!l!k!}\left(2x\right)^{k}\left.\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\tau^{2n+k}t^{2l+k}\right|_{t,\tau=0}$
$\displaystyle=2^{m}m!\sum_{n=0}^{\left[m/2\right]}\frac{m!}{2^{2n}\left(n!\right)^{2}\left(m-2n\right)!}x^{m-2n},$
(A5)

and then comparing Eq.(A5) with Eq.(A1) we can obtain Eq.(7).

APPENDIX B: Derivation of Wigner function Eq.(31) of PASV

In this appendix, we present the details for deriving the Wigner function
Eq.(31). Substituting Eq.(23) into Eq.(30) and noticing Eq.(19) as well as the
generating function of single-variable Hermite polynomials (5), we have

$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\bar{N}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\int\frac{d^{2}\alpha}{\pi}e^{-2\frac{\allowbreak\left|\zeta-\alpha
e^{-\kappa
t}\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}-2\left|\alpha\cosh
r-\allowbreak\alpha^{\ast}\sinh r\right|^{2}}$
$\displaystyle\times\left|H_{m-l}(-i\left(\alpha\cosh
r-\allowbreak\alpha^{\ast}\sinh r\right)\sqrt{2\coth r})\right|^{2}$
$\displaystyle=\bar{N}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}e^{-\upsilon^{2}-\tau^{2}}$
$\displaystyle\times\int\frac{d^{2}\alpha}{\pi}\left.\exp\left\\{-A\left|\alpha\right|^{2}+B_{1}\alpha+B_{2}\alpha^{\ast}+\left(\allowbreak\alpha^{2}+\alpha^{\ast}{}^{2}\right)\sinh
2r\right\\}\right|_{\tau=\upsilon=0},$ (B1)

where $A$ and $B$ are given by Eqs.(34)-(35),

$\bar{N}=\frac{2\sinh^{m}r}{\pi\left(2\bar{n}+1\right)T}\frac{m!}{2^{m}P_{m}\left(\cosh
r\right)},$ (B2)

and

$\displaystyle B_{1}$ $\displaystyle=B^{\ast}\allowbreak-i2\left(\upsilon\cosh
r+\tau\sinh r\right)\sqrt{2\coth r},$ (B3) $\displaystyle B_{2}$
$\displaystyle=B+i2\allowbreak\left(\upsilon\sinh r+\tau\cosh
r\right)\sqrt{2\coth r}.$ (B4)

Further using the following integral identity [37]:

$\displaystyle\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi
z+\eta z^{\ast}+fz^{2}+gz^{\ast 2}\right)$
$\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left[\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right],$
(B5)

whose convergent condition is Re$\left(\zeta\pm f\pm g\right)<0,\ $and
Re$\left(\frac{\zeta^{2}-4fg}{\zeta\pm f\pm g}\right)<0,$ we can put Eq.(B1)
into the following form:

$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}$
$\displaystyle\times\exp\left\\{-\upsilon^{2}-\tau^{2}+\frac{A}{C}B_{1}B_{2}+\frac{\allowbreak
1}{C}\left(B_{1}^{2}+B_{2}^{2}\right)\sinh 2r\right\\}_{\tau=\upsilon=0},$
(B6)

where $C$ and $D$ are given by (34)-(35), and

$\displaystyle B_{1}B_{2}$ $\displaystyle=B^{\ast}B+2\left(\allowbreak\upsilon
D+\tau
D^{\ast}\right)+8\left(\tau^{2}+\upsilon^{2}\right)\cosh^{2}r+8\tau\upsilon\cosh
2r\coth r,$ (B7) $\displaystyle B_{1}^{2}+B_{2}^{2}$
$\displaystyle=B^{\ast}{}^{2}+B{}^{2}-32\tau\upsilon\cosh^{2}r-8\left(\upsilon^{2}+\tau^{2}\right)\cosh
2r\coth r-\allowbreak 4\tau D-4\upsilon D^{\ast},$ (B8)

then substituting Eqs.(B7)-(B8) into Eq.(B6) yields

$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}e^{\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}$
$\displaystyle\times\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}\exp\left\\{-G\left(\upsilon^{2}+\tau^{2}\right)+\allowbreak
2\upsilon E+2\tau E^{\ast}+\tau\upsilon F\right\\}_{\tau=\upsilon=0},$ (B9)

where $E,F,$ and $G$ are given by Eqs.(35)-(37). In order to further simplify
Eq.(B9), expanding the exponential item $e^{\tau\upsilon F}$ as sum series and
using Eqs.(5) and the formula

$\left.\frac{\partial^{m}}{\partial\upsilon^{m}}e^{-G\upsilon^{2}+\allowbreak
2\upsilon E}\right|_{\upsilon=0}=G^{m/2}H_{m}\left(E/\sqrt{G}\right),$ (B10)

we see

$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}+\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{F^{k}}{k!}\frac{\partial^{2k}}{\partial\left(2E\right)^{k}\partial\left(2E^{\ast}\right)^{k}}\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}\left.e^{-G\left(\upsilon^{2}+\tau^{2}\right)+\allowbreak
2\upsilon E+2\tau E^{\ast}}\right|_{\tau=\upsilon=0}$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}+\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}G^{m-l}}{l!\left[\left(m-l\right)!\right]^{2}}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{F^{k}}{k!}\frac{\partial^{2k}}{\partial\left(2E\right)^{k}\partial\left(2E^{\ast}\right)^{k}}\left|H_{m-l}\left(E/\sqrt{G}\right)\right|^{2}.$
(B11)

Then using Eq.(22) yields Eq.(31).

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