# Multi-photon-addition amplified coherent state

Xue-feng Zhan1, Qiang Ke1, Min-xiang Li2, and Xue-xiang Xu1,† 1College of
Physics and Communication Electronics, Jiangxi Normal University, Nanchang
330022, China;
2School of Education, Jiangxi Normal University, Nanchang 330022, China;
<EMAIL_ADDRESS>
###### Abstract

State $g^{\hat{n}}\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$ and state
$\hat{a}^{{\dagger}m}g^{\hat{n}}\left|\alpha\right\rangle$ are same to state
$\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$, which is called as multi-
photon-addition amplified coherent state (MPAACS) by us. Here, $\hat{n}$,
$\hat{a}^{{\dagger}}$, $\left|\alpha\right\rangle$, $g$ ($\geq 1$), and $m$
are photon number operator, creation operator, coherent state, gain facor, and
an interger, respectively. We study mathematical and physical properties for
these MPAACSs, including normalization, photon component analysis, Wigner
function, effective gain, quadrature squeezing, and equivalent input noise.
Actually, the MPAACS, which contains more nonclassicality, is an amplified
version of photon-added coherent state (PACS) introduced by Agrwal and Tara
[Phys. Rev. A 43, 492 (1991)]. Our work provides theoretical references for
implementing amplifiers for light fields.

Keywords: noiseless linear amplification; repeated photon addition; amplified
coherent state; noise; Wigner function

##  I Introduction

In physics, signal amplification is a simple concept. However, signal
amplification unavoidably comes with noise1 . Often, the introduced noise
makes it difficult for people to distinguish the amplified signal. The
downside is that quantum noise will restrict quantum technologies such as
quantum cloning2 ; 3  and superluminal information transfer4 . In order to
conquer this restriction, people are more looking forward to taking advantage
of noiseless amplification, which can be implemented by probabilistic
operations5 ; 6 . In recent years, the theoretical and experimental study on
ideal noiseless linear amplification (NLA) has attracted the interests of the
researchers7 ; 8 ; 9 ; 10 . The NLA has been used for many quantum information
tasks such as loss suppression11 , quantm repeater12 , quantum error
correction13 , and entanglement distillation14 .

The NLA can be described by the operator $g^{\hat{n}}$ (AM), where
$\hat{n}=\hat{a}^{{\dagger}}\hat{a}$ is the photon number operator and $g$
($g>1$) is the gain factor. Here $g^{\hat{n}}\equiv\hat{1}$ is just the
identity operator if $g=1$. The NLA can be implemented probabilistically by
combinating multiple photon addition and subtraction with current technology15
. Theoretically, an ideal noiseless amplifier (described by $g^{\hat{n}}$) can
map an input state $\rho_{in}$ into an output state $\rho_{out}$, i.e.,
$g^{\hat{n}}:\rho_{in}\longmapsto\rho_{out}$. For input Fock state
$\left|n\right\rangle$, we have
$g^{\hat{n}}:\left|n\right\rangle\longmapsto\left|n\right\rangle$ due to
$\hat{n}\left|n\right\rangle=n\left|n\right\rangle$. For input coherent state
(CS), we have
$g^{\hat{n}}:\left|\alpha\right\rangle\longmapsto\left|g\alpha\right\rangle$
because of
$g^{\hat{n}}\left|\alpha\right\rangle=e^{(g^{2}-1)\left|\alpha\right|^{2}/2}\left|g\alpha\right\rangle$.
Generally, the input states to be amplified by $g^{\hat{n}}$ include Gaussian
and non-Gaussian state16 . Moreover, the output states must be re-normalized
after operating $g^{\hat{n}}$ because this operator is unbounded.

In 2016, Park et al. suggested and compared several schemes for non-
deterministic NLA of CSs using $\hat{a}^{{\dagger}2}$,
$\hat{a}\hat{a}^{{\dagger}}$, $\left(\hat{a}\hat{a}^{{\dagger}}\right)^{2}$,
$\hat{a}^{{\dagger}4}$, $\hat{a}\hat{a}^{{\dagger}}\hat{a}^{{\dagger}2}$,
$\hat{a}^{{\dagger}2}\hat{a}\hat{a}^{{\dagger}}$, which may work as amplifiers
for CSs with weak, medium or large amplitudes. Among them, the two-photon
addition ($\hat{a}^{{\dagger}2}$) scheme work more effectively than others as
a noiseless amplifier17 . Before then, in 1991, Argarwal and Tara had
introduced photon-added CSs (PACS)
$\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$, which can be generated by
applying a $m$-photon addition $\hat{a}^{{\dagger}m}$ (AD) on the CS
$\left|\alpha\right\rangle$, where $m$ is an interger18 . It is undeniable
that the AD $\hat{a}^{{\dagger}m}$ also works as an amplifier to amplify
$\left|\alpha\right\rangle$ to
$\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$.

In this paper, we shall study the behaviour of
$\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$ under the action of
$g^{\hat{n}}$. In another words, we shall obtain new quantum states by further
applying $g^{\hat{n}}$ on $\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$. We
will analyze their nonclassical properties and amplification effects. The key
is to examine the combinatorial effects of AD $\hat{a}^{{\dagger}m}$ and AM
$g^{\hat{n}}$ on CSs. The remaining paper is organized as follows. In Sec.2,
we introduce our considered states accompanying with description and
normalization. In Sec.3, we analyze photon components for them. In Sec.4, we
study Wigner functions to show character of Gaussianity and nonclassicality.
Sec.5 is devoted to studying effective gain, quadrature squeezing and
equivalent input noise for these states. Conclusions are summarized in the
last section.

##  II Generating quantum states

Based on operators $g^{\hat{n}}$, $\hat{a}^{{\dagger}m}$, and CS
$\left|\alpha\right\rangle$, we introduce a class of new states by two
equivalent ways. The conceptual generating schemes are shown in Fig.1.

Figure 1: (a) Conceptional generating schemes of ADAMCS
$\left|\psi_{1}\right\rangle$ and AMADCS $\left|\psi_{2}\right\rangle$. It is
interesting to prove that $\left|\psi_{1}\right\rangle$ and
$\left|\psi_{2}\right\rangle$ are the same MPAACS $\left|\psi\right\rangle$.
(b) The map $\left|\alpha\right\rangle\longmapsto
a^{{\dagger}m}\left|g\alpha\right\rangle$ can be implemented by way 1
$\left|\alpha\right\rangle\longmapsto
a^{{\dagger}m}\left|\alpha\right\rangle\longmapsto
a^{{\dagger}m}\left|g\alpha\right\rangle$ or by way 2
$\left|\alpha\right\rangle\longmapsto\left|g\alpha\right\rangle\longmapsto
a^{{\dagger}m}\left|g\alpha\right\rangle$ after employing $g^{\hat{n}}$ and
$\hat{a}^{{\dagger}m}$ in sequence.

Way 1: Employing $\hat{a}^{{\dagger}m}$ then $g^{\hat{n}}$ on
$\left|\alpha\right\rangle$, we get the addition-amplification coherent state
(ADAMCS)

$\left|\psi_{1}\right\rangle=\frac{1}{\sqrt{N_{1}}}g^{\hat{n}}\hat{a}^{{\dagger}m}\left|\alpha\right\rangle,$
(1)

with normalization factor

$N_{1}=g^{2m}m!e^{\left(g^{2}-1\right)\left|\alpha\right|^{2}}L_{m}(-g^{2}\left|\alpha\right|^{2}),$
(2)

where $L_{m}(x)$ is the $m$th-order Laguerre polynomial19 . This map include
two processes, i.e., $\hat{a}^{{\dagger}m}:$
$\left|\alpha\right\rangle\longmapsto\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$
and $g^{\hat{n}}:$
$\hat{a}^{{\dagger}m}\left|\alpha\right\rangle\longmapsto\left|\psi_{1}\right\rangle$,
where $\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$ is the intermediate
state.

Way 2: Employing $g^{\hat{n}}$ then $\hat{a}^{{\dagger}m}$ on
$\left|\alpha\right\rangle$, we get the amplification-addition coherent state
(AMADCS)

$\left|\psi_{2}\right\rangle=\frac{1}{\sqrt{N_{2}}}\hat{a}^{{\dagger}m}g^{\hat{n}}\left|\alpha\right\rangle,$
(3)

with normalization factor

$N_{2}=m!e^{\left(g^{2}-1\right)\left|\alpha\right|^{2}}L_{m}(-g^{2}\left|\alpha\right|^{2}).$
(4)

This map include two processes, i.e., $g^{\hat{n}}:$
$\left|\alpha\right\rangle\longmapsto\left|g\alpha\right\rangle$ and
$\hat{a}^{{\dagger}m}:$
$\left|g\alpha\right\rangle\longmapsto\left|\psi_{2}\right\rangle$, where
$\left|g\alpha\right\rangle$ is the intermediate state.

Although operators $g^{\hat{n}}$ and $\hat{a}^{{\dagger}m}$ do not commute
with each other (i.e.
$g^{\hat{n}}\hat{a}^{{\dagger}m}\neq\hat{a}^{{\dagger}m}g^{\hat{n}}$), the
relation
$g^{\hat{n}}\hat{a}^{{\dagger}m}=g^{m}\hat{a}^{{\dagger}m}g^{\hat{n}}$ is
satisfied and leads to
$\left|\psi_{1}\right\rangle=\left|\psi_{2}\right\rangle$ after normalization.
Thus, we redefine them as $\left|\psi\right\rangle$ in the form

$\left|\psi\right\rangle=\frac{1}{\sqrt{N}}\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle,$
(5)

with normalization factor $N=m!L_{m}(-g^{2}\left|\alpha\right|^{2})$. In this
paper, we call $\left|\psi\right\rangle$ as MPAACS, which is just an amplified
PACS. By using $\left|\psi_{1}\right\rangle$ in Appendix A or by using
$\left|\psi_{2}\right\rangle$ in Appendix B in two parallel ways, we get the
expressions of state description, normalization facor, expectation value,
density matrix elements, and Wigner function for $\left|\psi\right\rangle$.
Without question, the results in Appendix A are same to those in Appendix B
except $N_{1}=g^{2m}N_{2}$. As illustrated in Table I, states including
$\left|0\right\rangle$, $\left|m\right\rangle$, $\left|\alpha\right\rangle$,
$\left|g\alpha\right\rangle$, and
$\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$ are special cases of
$\left|\psi\right\rangle$ with proper $g$, $\alpha$, and $m$.

Table 1: Special cases of the MPAACS $\left|\psi\right\rangle$ | parameter conditions | state name
---|---|---
$\left|0\right\rangle$ | $m=0,\alpha=0$ | vacuum state
$\left|m\right\rangle$ | $\alpha=0$ | Fock state
$\left|\alpha\right\rangle$ | $g=1,m=0$ | coherent state
$\left|g\alpha\right\rangle$ | $m=0$ | amplified CS
$a^{{\dagger}m}\left|\alpha\right\rangle$ | $g=1$ | photon-added CS

##  III Component analysis of the MPAACS

The MPAACS can be expanded in terms of Fock states as

$\left|\psi\right\rangle=\sum_{k=m}^{\infty}c_{k}\left|k\right\rangle,$ (6)

with

$c_{k}=\frac{\sqrt{k!}\left(g\alpha\right)^{k-m}e^{-g^{2}\left|\alpha\right|^{2}/2}}{\left(k-m\right)!\sqrt{m!L_{m}(-g^{2}\left|\alpha\right|^{2})}}.$
(7)

It is interesting to note that the components including $\left|k\right\rangle$
($k<m$) are missing in the MPAACS. Moreover, when $g=1$, the result can be
reduced to that in the work of Agarwal and Tara18 . Accordingly, the density
matrix element (DME) for $\rho=\left|\psi\right\rangle\left\langle\psi\right|$
can be expressed as $\rho_{kl}=\left\langle
k\right|\rho\left|l\right\rangle=c_{k}c_{l}^{\ast}$, whose numerical results
also can be calculated according to Eq.(A.7) or Eq.(B.6) in the Appendix. The
photon number distribution (PND) can be written as
$\rho_{kk}=\left|c_{k}\right|^{2}$, i.e., the diagonal terms of the density
matrix. Without loss of generality, we shall take
$\alpha=\left|\alpha\right|e^{i\theta_{p}}$ only with $\theta_{p}=0$ in the
numerical analysis.

Fig.2 presents the PNDs $\rho_{kk}$ of the MPAACSs with different parameters
($\left|\alpha\right|$, $g$, $m$). The results show that: (a) The effect of
photon-added number $m$ can be observed in Fig.2(a) at fixed
$\left|\alpha\right|=1$, $g=2$ and for three different $m$. As $m$ increases,
the PNDs $\rho_{kk}$ approach higher-photon regime where all the
$\left|k\right\rangle$ terms with $k<m$ are missing, like a shifted version of
original amplified CS ($m=0$). (b) The effect of gain factor $g$ can be
observed in Fig.2(b) at fixed $\left|\alpha\right|=1$, $m=2$ and for three
different $g$. It is obvious to see that the PNDs approach higher-photon
regime as $g$ increasing. (c) The effect of the field amplitude
$\left|\alpha\right|$ can be observed in Fig.2(c) at fixed $g=2$, $m=2$ and
for three different $\left|\alpha\right|$. We find if $\left|\alpha\right|=0$,
there is the sole component of $\left|m\right\rangle$. If case
$\left|\alpha\right|>0$, the PNDs approach higher-photon regime as
$\left|\alpha\right|$ increasing.

Fig.3 presents the absolute values of the DMEs ($\left|\rho_{kl}\right|$) of
the MPAACSs showing the effects of parameters ($\left|\alpha\right|$, $g$,
$m$). Generally, two adjacent states, corresponding to two adjacent graphs in
Fig.3, can be connected through a map with its operation. These maps include
$\hat{a}^{{\dagger}m}:\left|0\right\rangle\longmapsto\left|m\right\rangle$ for
(a)-(b);
$D\left(\alpha\right):\left|0\right\rangle\longmapsto\left|\alpha\right\rangle$
for (a)-(c), where $D\left(\alpha\right)=e^{\alpha
a^{{\dagger}}-\alpha^{\ast}a}$ is the displacement operator;
$\hat{a}^{{\dagger}m}:\left|\alpha\right\rangle\longmapsto\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$
for (c)-(d);
$g^{\hat{n}}:\left|\alpha\right\rangle\longmapsto\left|g\alpha\right\rangle$
for (c)-(e);
$g^{\hat{n}}:\hat{a}^{{\dagger}m}\left|\alpha\right\rangle\longmapsto\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$
for (d)-(f);
$\hat{a}^{{\dagger}m}:\left|g\alpha\right\rangle\longmapsto\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$
for (e)-(f). Of course, the map
$\left|m\right\rangle\longmapsto\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$
for (b)-(d) can not be realized simply by $D\left(\alpha\right)$. It can be
easily seen from Fig.3 that: (1) The AD $\hat{a}^{{\dagger}m}$ leads to the
rescale of corrresponding DMEs and the displacement to higher indeces
($\rho_{k,l}\rightarrow\rho_{k+m,l+m}$), leaving all $\rho_{k,l}$ with $k,l<m$
void. (2) The AM $g^{\hat{n}}$ leads to the rescale of corrresponding DMEs.

Figure 2: PNDs of $\left|\psi\right\rangle$ with different
($\left|\alpha\right|$, $g$, $m$). (a) showing the effect of $m$; (b) showing
the effect of $g$; (c) showing the effect of $\left|\alpha\right|$. Figure 3:
Absolute values of DMEs for $\left|\psi\right\rangle$-($\left|\alpha\right|$,
$g$, $m$) taking (a) $\left|0\right\rangle$-($0$, $1$, $0$); (b)
$\left|m\right\rangle$-($0$, $1$, $2$); (c) $\left|\alpha\right\rangle$-($1$,
$1$, $0$); (d) $a^{{\dagger}m}\left|\alpha\right\rangle$-($1$, $1$, $2$); (e)
$\left|g\alpha\right\rangle$-($1$, $2$, $0$); and (f)
$a^{{\dagger}m}\left|g\alpha\right\rangle$-($1$, $2$, $2$). Indeed, any two
adjacent states can be connected through appropriate operations such as
$D\left(\alpha\right)$, $a^{{\dagger}m}$, and $g^{\hat{n}}$.

##  IV Wigner function of the MPAACS

In phase-space formalism, the Wigner function (WF) is an important
quasiprobability distribution, representing the corresponding quantum state20
; 21 ; 22 . One can judge Gaussianity or non-Gaussianity from its WF form and
non-classicality from its Wigner negativity23 ; 24 . Theoretically,
$W_{\rho}\left(\beta\right)$ can be also obtained by means of the following
transformation

$W_{\rho}\left(\beta\right)=\sum_{k,l=0}^{\infty}\rho_{kl}W_{\left|k\right\rangle\left\langle
l\right|}\left(\beta\right)$ (8)

which is associated with $\rho_{kl}$ and $W_{\left|k\right\rangle\left\langle
l\right|}\left(\beta\right)$ (WF of operator $\left|k\right\rangle\left\langle
l\right|$). But it is very difficult to simulate perfectly as given by Eq.(8)
because of the infinite summation. In experiment, the approximate WF can be
reconstructed from a truncated density matrix with finite dimension through
tomographic analysis.

Fortunately, analytical WF of the MPAACS can be obtained from Eq.(A.8) or
Eq.(B.7) as follows

$W_{\rho}\left(\beta\right)=\dfrac{2(-1)^{m}L_{m}(\left|2\beta-g\alpha\right|^{2})}{\pi
L_{m}(-g^{2}\left|\alpha\right|^{2})}e^{-2\left|\beta-g\alpha\right|^{2}},$
(9)

in the ($x,y$) phase space, where $\beta=(x+iy)/\sqrt{2}$. As expected, Eq.(9)
can reduce to the following special cases.

(1) If $\alpha=0$ and $m=0$, then we obtain WF of vacuum state

$W_{\left|0\right\rangle}\left(\beta\right)=\dfrac{2}{\pi}e^{-2\left|\beta\right|^{2}}.$
(10)

(2) If $m=0$ and $g=1$, then we obtain WF of coherent state

$W_{\left|\alpha\right\rangle}\left(\beta\right)=\dfrac{2}{\pi}e^{-2\left|\beta-\alpha\right|^{2}}.$
(11)

(3) If $m=0$, then we obtain WF of amplified CS

$W_{\left|g\alpha\right\rangle}\left(\beta\right)=\dfrac{2}{\pi}e^{-2\left|\beta-g\alpha\right|^{2}}.$
(12)

(4) If $\alpha=0$, then we obtain WF of Fock state

$W_{\left|m\right\rangle}\left(\beta\right)=\dfrac{2}{\pi}(-1)^{m}e^{-2\left|\beta\right|^{2}}L_{m}(4\left|\beta\right|^{2}).$
(13)

(5) If $g=1$, then we obtain WF of photon-added CS

$W_{\hat{a}^{{\dagger}m}\left|\alpha\right\rangle}\left(\beta\right)=\dfrac{2(-1)^{m}L_{m}(\left|2\beta-\alpha\right|^{2})}{\pi
L_{m}(-\left|\alpha\right|^{2})}e^{-2\left|\beta-\alpha\right|^{2}}.$ (14)

Eq.(14) is just the equation (3.8) in Ref.18 .

Corresponding to Fig.3 and according to Eqs.(9)-(14), we plot WFs in Fig.4 and
their sections (with $y=0$) in Fig.5(a), as well as their marginal
distributions in Fig.5(b). Here, the marginal distribution in $x$ direction
can be evaluated numerically by

$p\left(x\right)=\int_{-\infty}^{\infty}W\left(\beta\right)dy,$ (15)

where the scaling relations such as
$\int_{-\infty}^{\infty}W\left(\beta\right)d^{2}\beta=1$ and
$\int_{-\infty}^{\infty}p\left(x\right)dx=1$ must be ensured.

Figure 4: WFs $W\left(\beta\right)$ of $\left|\psi\right\rangle$ in the phase
space $(x,y)$. Here, $\left|\psi\right\rangle$-($\left|\alpha\right|$, $g$,
$m$) are taking (a) $\left|0\right\rangle$-($0$, $1$, $0$); (b)
$\left|m\right\rangle$-($0$, $1$, $2$); (c) $\left|\alpha\right\rangle$-($1$,
$1$, $0$); (d) $a^{{\dagger}m}\left|\alpha\right\rangle$-($1$, $1$, $2$); (e)
$\left|g\alpha\right\rangle$-($1$, $2$, $0$); and (f)
$a^{{\dagger}m}\left|g\alpha\right\rangle$-($1$, $2$, $2$). Figure 5: (a)
Sections $W(x,y=0)$ and (b) Marginal distribution $p(x)$ of
$W\left(\beta\right)$ for $\left|\psi\right\rangle$ in Fig.4 as a function of
$x$. Here, $\left|\psi\right\rangle$-($\left|\alpha\right|$, $g$, $m$) taking
(red dashed) $\left|0\right\rangle$-($0$, $1$, $0$); (red solid)
$\left|m\right\rangle$-($0$, $1$, $2$); (blue dashed)
$\left|\alpha\right\rangle$-($1$, $1$, $0$); (blue solid)
$a^{{\dagger}m}\left|\alpha\right\rangle$-($1$, $1$, $2$); (black dashed)
$\left|g\alpha\right\rangle$-($1$, $2$, $0$); and (black solid)
$a^{{\dagger}m}\left|g\alpha\right\rangle$-($1$, $2$, $2$).

The left three figures of Fig.4 correspond to vacuum state
$\left|0\right\rangle$, coherent state $\left|\alpha\right\rangle$, and
amplified CS $\left|g\alpha\right\rangle$. Their distributions are Gaussian
without Wigner negativity and with different central positions. The right
three figures of Fig.4 correspond to Fock state $\left|m\right\rangle$,
Photon-added CS $\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$, and MPAACS
$\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$. Their distributions are non-
Gaussian and showing Wigner negativity. Main characters including Gaussianity
(or non-Gaussianity) and nonclassicality (negativity) can also be seen from
their corresponding sections in $y=0$ in Fig.5(a). Morover, we see from
Fig.5(b) that, as $m$ is increased, the width of distribution $p(x)$ becomes
narrower comparing to that of $\left|\alpha\right\rangle$ and
$\left|g\alpha\right\rangle$ (corresponding to $m=0$) with
$\left|\alpha\right|>0$. We think, this is owe to the effect of AD
$\hat{a}^{{\dagger}m}$.

##  V Effective gain, quadrature squeezing and equivalent input noise

Generally, many properties of light states are related to amplitude quadrature
$\hat{x}=\left(a+a^{{\dagger}}\right)/\sqrt{2}$ and phase quadrature
$\hat{p}=\left(a-a^{{\dagger}}\right)/(i\sqrt{2})$25 ; 26 . In order to study
the combinatorial contributions of AM $g^{\hat{n}}$ and AD
$\hat{a}^{{\dagger}m}$, we compare the properties of
$\left|\alpha\right\rangle$ and $\left|\psi\right\rangle$ in terms of the
effective gain, quadrature squeezing, and equivalent input noise. After our
full consideration, we only use quadrature operator $\hat{x}$ to discuss all
these properties.

Effective gain: An effective gain27 ; 28  from the input
$\left|\alpha\right\rangle$ to the output $\left|\psi\right\rangle$ can be
defined as the ratio of the expectation values of the quadrature operator
$\hat{x}$:

$g_{eff}=\frac{\left\langle\hat{x}\right\rangle_{\left|\psi\right\rangle}}{\left\langle\hat{x}\right\rangle_{\left|\alpha\right\rangle}}.$
(16)

In particularly, we have

$\displaystyle g_{eff}^{(m=0)}$ $\displaystyle=$ $\displaystyle g,$
$\displaystyle g_{eff}^{(m=1)}$ $\displaystyle=$ $\displaystyle
g\frac{2+g^{2}\left|\alpha\right|^{2}}{1+g^{2}\left|\alpha\right|^{2}},$
$\displaystyle g_{eff}^{(m=2)}$ $\displaystyle=$ $\displaystyle
g\frac{6+6g^{2}\left|\alpha\right|^{2}+g^{4}\left|\alpha\right|^{4}}{2+4g^{2}\left|\alpha\right|^{2}+g^{4}\left|\alpha\right|^{4}},$
(17) $\displaystyle\vdots$

Figure 6: (a) Effective gain $g_{eff}$, (b) Variances
$\left\langle\left(\Delta x\right)^{2}\right\rangle$, and (c) Equivalent input
noise $N_{eq}$ as the function of $\left|\alpha\right|$ . Here, parameters are
taking $g=1$ (red)$,2$ (blue)$,3$ (black) and $m=0$ (dashed)$,1$
(dotdashed)$,2$ (solid).

In Fig.6(a), we plot $g_{eff}$ as a function of $\left|\alpha\right|$ by
taking different $g$ and $m$. From Fig.6(a), we find that $g_{eff}\gtrsim g$
is always right and $g_{eff}$ is a monotonical decreasing function of
$\left|\alpha\right|$ for different $g$ and $m>0$. Two limiting cases include:
(1) $g_{eff}\rightarrow(m+1)g$ in the limit of $\left|\alpha\right|\rightarrow
0$; (2) $g_{eff}\rightarrow g$ in the limit of
$\left|\alpha\right|\rightarrow\infty$.

Quadrature squeezing: Quadrature squeezing, as a nonclassical character of
light states, can be evident by measuring the quadrature variances29 ; 30 .
The quadrature variance in $\hat{x}$ can be calculated from

$\left\langle\left(\Delta\hat{x}\right)^{2}\right\rangle=\left\langle\hat{x}^{2}\right\rangle-\left\langle\hat{x}\right\rangle^{2},$
(18)

which is also called as the quadrature flunctuation. Similar definition
$\left\langle\left(\Delta\hat{p}\right)^{2}\right\rangle$ is available to
quadrature $\hat{p}$. It is well known that $\left|\alpha\right\rangle$ and
$\left|g\alpha\right\rangle$ are not quadrature squeezing states because of
$\left\langle\left(\Delta\hat{x}\right)^{2}\right\rangle=\left\langle\left(\Delta\hat{p}\right)^{2}\right\rangle=0.5$.
While if $\left\langle\left(\Delta\hat{x}\right)^{2}\right\rangle$ or
$\left\langle\left(\Delta\hat{p}\right)^{2}\right\rangle$ is smaller than
$0.5$, then the quantum state is quadrature squeezing. Thus, a question
naturally arises: is the MPAACS quadrature squeezing? In order to answer this
question, we only plot
$\left\langle\left(\Delta\hat{x}\right)^{2}\right\rangle$ as a function of
$\left|\alpha\right|$ for different $\left|\psi\right\rangle$ in Fig.6(b)
regardless of $\left\langle\left(\Delta\hat{p}\right)^{2}\right\rangle\geq
0.5$.

Clearly, we find that (1) For $m=0$ case, the $\hat{x}$ quadrature variance
remains constant fluctuation $0.5$ (without quadrature squeezing) for any $g$
and $\left|\alpha\right|$. This is because $\left|\psi\right\rangle$ has been
reduced to $\left|\alpha\right\rangle$ or $\left|g\alpha\right\rangle$ in this
case. (2) For $m=1$ case, the $\hat{x}$ quadrature exhibits squeezing if
$\left|g\alpha\right|>1$, which is consistent with the result for SPACS31 .
(3) For $m=2$ case, the $\hat{x}$ quadrature exhibits squeezing if
$\left|g\alpha\right|>0.938744$. Other similar reduced fluctuations will be
exhibited squeezing if $\left|g\alpha\right|>0.900407$ for $m=3$,
$\left|g\alpha\right|>0.873904$ for $m=4$, $\left|g\alpha\right|>0.854454$ for
$m=5$, and so on. (4) All above results show that the MPAACSs except $m=0$
will exhibit quadrature squeezing, only when $\left|g\alpha\right|$ exceeds a
certain threshold. (5) Moreover, we always have
$\left\langle\left(\Delta\hat{x}\right)^{2}\right\rangle\rightarrow 0.5$ in
the limit of $\left|\alpha\right|\rightarrow\infty$.

Equivalent input noise: Noise of the output $\left|\psi\right\rangle$
referring to the input $\left|\alpha\right\rangle$ may be analyzed in terms of
the equivalent input noise (EIN):

$N_{eq}=\frac{\left\langle\left(\Delta\hat{x}\right)^{2}\right\rangle_{\left|\psi\right\rangle}}{g_{eff}^{2}}-\left\langle\left(\Delta\hat{x}\right)^{2}\right\rangle_{\left|\alpha\right\rangle},$
(19)

which tells how much noise has been added to the input noise level. In fact,
the EIN came from a classical electronics terminology and used to quantify the
performance of an amplifier27 ; 28 ; 32 ; 33 . For an amplification process,
the EIN is negative and indicates the characteristic of noiseless
amplification34 .

In Fig.6(c), the EINs are shown as a function of $\left|\alpha\right|$ for
$\left|\psi\right\rangle$ with different $g$ and $m$. The numerical results
reveal that $N_{eq}$ is clearly negative for all $\left|\alpha\right|$, except
case $m=0$ and $g=1$. The main results include: (1) For $m=0$, we know that
$N_{eq}$ remains constant $0.5/g^{2}$ $-0.5$ for any $g$ and all
$\left|\alpha\right|$; (2) In the limit of $\left|\alpha\right|\rightarrow 0$,
we see $N_{eq}\rightarrow 0.5/g^{2}$ $-0.5$ for $m=0$, $0.375/g^{2}$ $-0.5$
for $m=1$, $0.277778/g^{2}$ $-0.5$ for $m=2$, respectively; (3) In the limit
of $\left|\alpha\right|\rightarrow\infty$, we see $N_{eq}\rightarrow$
$0.5/g^{2}$ $-0.5$ for any $g$ ($>1$) and different $m$; (4) A minimum value
of $N_{eq}$ can be see at proper $\left|\alpha\right|$ for each case.

##  VI Conclusion and discussion

We have introduced MPAACSs by applying AM $g^{\hat{n}}$ and AD
$\hat{a}^{{\dagger}m}$ on $\left|\alpha\right\rangle$ and proved that state
$g^{\hat{n}}\hat{a}^{{\dagger}m}\left|\alpha\right\rangle$ and state
$\hat{a}^{{\dagger}m}g^{\hat{n}}\left|\alpha\right\rangle$ are state
$\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$. From
$\left|\alpha\right\rangle$ to
$\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$, the combinatorial effect of
$g^{\hat{n}}$ and $\hat{a}^{{\dagger}m}$ is working as an amplifier. From the
point of view of quantum state engineering, these MPAACSs
$\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$ are a class of new quantum
states, which include many familiar quantum states, such as
$\left|0\right\rangle$, $\left|m\right\rangle$, $\left|\alpha\right\rangle$,
$\left|g\alpha\right\rangle$, and
$\hat{a}^{{\dagger}m}\left|g\alpha\right\rangle$. We have derived the
normalization factor for the MPAACS and found that it is related to Lagurrel
polynomials. Interesting physical properties are given analytically and
simulated numerically according to the supplementary materials. The main
results are summarized as follows.

As for the effects of AM $g^{\hat{n}}$ and AD $\hat{a}^{{\dagger}m}$ on photon
components of the MPAACSs, we find that: (1) The AD $\hat{a}^{{\dagger}m}$
leads to the void of the low-photon components (including
$\left|0\right\rangle$, $\left|1\right\rangle$, $\cdots$,
$\left|m-1\right\rangle$) and the re-layout of photon components. (2) The AM
$g^{\hat{n}}$ leads to the re-layout of photon components. As for the effects
of AM $g^{\hat{n}}$ and AD $\hat{a}^{{\dagger}m}$ on WFs, we find that: (1)
The AD $\hat{a}^{{\dagger}m}$ is a non-Gaussian operation, which can transform
a Gaussian state into a non-gaussian state, accompanying with Wigner
negativity. (2) The AM $g^{\hat{n}}$ is a Gaussian operation, which can remain
original Gaussianity or non-Gaussianity of quantum state. As for the effects
of AM $g^{\hat{n}}$ and AD $\hat{a}^{{\dagger}m}$ on amplification, squeezing
and noise, we find that: (1) Both $g^{\hat{n}}$ and $\hat{a}^{{\dagger}m}$ can
improve the effective gain by changing $g$ and $m$. (2) The quadrature
squeezing will exhibit when $\left|g\alpha\right|$ exceeds a certain threshold
except $m=0$. (3) The EINs except case $g=1$ and $m=0$ are negative, showing
the characteristic of noiseless amplification.

In our previous works35 ; 36 ; 37 , we have introduced several amplified
quantum states, such as amplified coherent state, amplified thermal state, and
amplified squeezed vacuum, by applying $\left(g-1\right)\hat{n}+1$ or
$(g-\sqrt{2g-1})\hat{n}^{2}+(\sqrt{2g-1}-1)\hat{n}+1$ on the corresponding
input states. These amplified states can exhibit their respective peculiar
nonclassicality. Of course, these operators work as amplifiers of realizing
signal amplification. However, it is actually impossible to implement a
perfect noiseless amplifier described by $g^{\hat{n}}$, albeit with zero
success probability38 . So, our present work only provides a theoretical
reference for signal amplification in quantum technology or state generation
in quantum state engineering.

## Appendix: Supplemental Materials

Using techniques such as $g^{\hat{n}}=:e^{(g-1)\hat{a}^{{\dagger}}\hat{a}}:$
($:\cdots:$ denotes normal ordering) and
$x^{n}=\partial_{s}^{n}e^{sx}|_{s=0}$, we provide the following supplemental
materials for all informations discussed in the main text.

### Appendix A: Information for state $\left|\psi_{1}\right\rangle$

In this appendix, we provide the state description, normalization, expectation
value, density matrix elements and Wigner function for
$\left|\psi_{1}\right\rangle$.

(a) State description

Eq.(1) can be further written as

$\left|\psi_{1}\right\rangle=\frac{e^{-\frac{\left|\alpha\right|^{2}}{2}}}{\sqrt{N_{1}}}\partial_{s_{1}}^{m}e^{g\left(\alpha+s_{1}\right)a^{{\dagger}}}\left|0\right\rangle|_{s_{1}=0},$
(A.1) accompanying with conjugate state
$\left\langle\psi_{1}\right|=\frac{e^{-\frac{\left|\alpha\right|^{2}}{2}}}{\sqrt{N_{1}}}\partial_{t_{1}}^{m}\left\langle
0\right|e^{g\left(\alpha^{\ast}+t_{1}\right)a}|_{t_{1}=0},$ (A.2) which leads
to density operator
$\rho_{1}=\left|\psi_{1}\right\rangle\left\langle\psi_{1}\right|$,
$\rho_{1}=\frac{e^{-\left|\alpha\right|^{2}}}{N_{1}}\partial_{s_{1}}^{m}\partial_{t_{1}}^{m}e^{g\left(\alpha+s_{1}\right)a^{{\dagger}}}\left|0\right\rangle\left\langle
0\right|e^{g\left(\alpha^{\ast}+t_{1}\right)a}|_{\left(s_{1},t_{1}\right)=0}.$
(A.3)

(b) Normalization

The normalization factor is

$N_{1}=e^{\left(g^{2}-1\right)\left|\alpha\right|^{2}}\partial_{s_{1}}^{m}\partial_{t_{1}}^{m}e^{g^{2}\left(t_{1}\alpha+s_{1}\alpha^{\ast}+\allowbreak
s_{1}t_{1}\right)}|_{\left(s_{1},t_{1}\right)=0}.$ (A.4) Using the following
formula
$L_{m}\left(xy\right)=\frac{(-1)^{m}}{m!}\partial_{s}^{m}\partial_{t}^{m}e^{-st+sx+ty}|_{\left(s,t\right)=0},$
(A.5)

we easily obtain the analytical result of $N_{1}$ in Eq.(2).

(c) Expectation value

Here, we give $\langle a^{\dagger k}a^{l}\rangle_{\rho_{1}}$ as follows

$\displaystyle\langle a^{\dagger k}a^{l}\rangle_{\rho_{1}}$
$\displaystyle=\frac{e^{\left(g^{2}-1\right)\left|\alpha\right|^{2}}}{N_{1}}\partial_{s_{1}}^{m}\partial_{t_{1}}^{m}\partial_{f_{1}}^{k}\partial_{h_{1}}^{l}$
$\displaystyle
e^{g\left(h_{1}\alpha+f_{1}\alpha^{\ast}\right)+g^{2}\left(t_{1}\alpha+s_{1}\alpha^{\ast}+\allowbreak
s_{1}t_{1}\right)\allowbreak}$ $\displaystyle
e^{g\left(h_{1}s_{1}+f_{1}t_{1}\right)}|_{\left(s_{1},t_{1},f_{1},h_{1}\right)=0}.$
(A.6)

(d) Density matrix elements

Here, we give $\rho_{kl}^{(1)}=\left\langle k|\rho_{1}|l\right\rangle$ with
the following form

$\displaystyle\rho_{kl}^{(1)}$
$\displaystyle=\frac{e^{-\left|\alpha\right|^{2}}}{N_{1}\sqrt{k!l!}}\partial_{s_{1}}^{m}\partial_{t_{1}}^{m}\partial_{f_{1}}^{k}\partial_{h_{1}}^{l}$
$\displaystyle
e^{gf_{1}\left(\alpha+s\right)+gh_{1}\left(\alpha^{\ast}+t\right)}|_{\left(s_{1},t_{1},f_{1},h_{1}\right)=0}.$
(A.7)

(e) Wigner function

WF of $\rho_{1}$ has the following form

$\displaystyle W_{\rho_{1}}\left(\beta\right)$ $\displaystyle=\dfrac{2}{\pi
N_{1}}e^{-\left(g^{2}+1\right)|\alpha|^{2}-2\left|\beta\right|^{2}+2g\allowbreak\alpha\beta^{\ast}+2g\alpha^{\ast}\beta}$
$\displaystyle\partial_{s_{1}}^{m}\partial_{t_{1}}^{m}e^{-g^{2}\left(t_{1}\alpha+s_{1}\alpha^{\ast}+s_{1}t_{1}\right)+\allowbreak
2g\left(t_{1}\beta+s_{1}\beta^{\ast}\right)}|_{\left(s_{1},t_{1}\right)=0}$
(A.8)

### Appendix B: Information for state $\left|\psi_{2}\right\rangle$

In this appendix, we provide the state description, normalization, expectation
value, density matrix elements and Wigner function for
$\left|\psi_{2}\right\rangle$.

(a) State description

Eq.(3) can be further written as

$\left|\psi_{2}\right\rangle=\frac{e^{-\frac{\left|\alpha\right|^{2}}{2}}}{\sqrt{N_{2}}}\partial_{s_{2}}^{m}e^{\left(s_{2}+g\alpha\right)a^{{\dagger}}}\left|0\right\rangle|_{s_{2}=0},$
(B.1) accompanying with conjugate state
$\left\langle\psi_{2}\right|=\frac{e^{-\frac{\left|\alpha\right|^{2}}{2}}}{\sqrt{N_{2}}}\partial_{t_{2}}^{m}\left\langle
0\right|e^{\left(t_{2}+g\alpha^{\ast}\right)a}|_{t_{2}=0},$ (B.2) and leads to
density operator
$\rho_{2}=\left|\psi_{2}\right\rangle\left\langle\psi_{2}\right|$,
$\rho_{2}=\frac{e^{-\left|\alpha\right|^{2}}}{N_{2}}\partial_{s_{2}}^{m}\partial_{t_{2}}^{m}e^{\left(s_{2}+g\alpha\right)a^{{\dagger}}}\left|0\right\rangle\left\langle
0\right|e^{\left(t_{2}+g\alpha^{\ast}\right)a}|_{\left(s_{2},t_{2}\right)=0}.$
(B.3)

(b) Normalization

The normalization factor is

$N_{2}=e^{\left(g^{2}-1\right)\left|\alpha\right|^{2}}\partial_{s_{2}}^{m}\partial_{t_{2}}^{m}e^{gt_{2}\alpha+gs_{2}\alpha^{\ast}\allowbreak+\allowbreak
s_{2}t_{2}}|_{\left(s_{2},t_{2}\right)=0},$ (B.4) which leads to the
analytical result of $N_{2}$ in Eq.(4).

(c) Expectation value

Here, we give $\langle a^{\dagger k}a^{l}\rangle_{\rho_{2}}$ as follows

$\displaystyle\langle a^{\dagger k}a^{l}\rangle_{\rho_{2}}$
$\displaystyle=\frac{e^{\left(g^{2}-1\right)\left|\alpha\right|^{2}}}{N_{2}}\partial_{s_{2}}^{m}\partial_{t_{2}}^{m}\partial_{f_{2}}^{k}\partial_{h_{2}}^{l}$
$\displaystyle
e^{g\left(h_{2}+t_{2}\right)\alpha+g\allowbreak\left(f_{2}+s_{2}\right)\alpha^{\ast}}$
$\displaystyle e^{\allowbreak h_{2}s_{2}+\allowbreak
f_{2}t_{2}+s_{2}t_{2}}|_{\left(s_{2},t_{2},f_{2},h_{2}\right)=0}.$ (B.5)

(d) Density matrix elements

Here, we give $\rho_{kl}^{(2)}=\left\langle k|\rho_{2}|l\right\rangle$ with
the following form

$\displaystyle\rho_{kl}^{(2)}$
$\displaystyle=\frac{e^{-\left|\alpha\right|^{2}}}{N_{2}\sqrt{k!l!}}\partial_{s_{2}}^{m}\partial_{t_{2}}^{m}\partial_{f_{2}}^{k}\partial_{h_{2}}^{l}$
$\displaystyle
e^{f_{2}s_{2}+h_{2}t_{2}+g\left(f_{2}\alpha+h_{2}\alpha^{\ast}\right)}|_{\left(s_{2},t_{2},f_{2},h_{2}\right)=0}.$
(B.6)

(e) Wigner function

WF of $\rho_{2}$ has the following form

$\displaystyle W_{\rho_{2}}\left(\beta\right)$ $\displaystyle=\dfrac{2}{\pi
N_{2}}e^{-\left(g^{2}+1\right)|\alpha|^{2}-2\left|\beta\right|^{2}+2g\allowbreak\left(\alpha\beta^{\ast}+\alpha^{\ast}\beta\right)}$
$\displaystyle\partial_{s_{2}}^{m}\partial_{t_{2}}^{m}e^{-g\left(t_{2}\alpha+s_{2}\alpha^{\ast}\right)-s_{2}t_{2}+\allowbreak
2\left(t_{2}\beta+s_{2}\beta^{\ast}\right)}|_{\left(s_{2},t_{2}\right)=0}$
(B.7)

###### Acknowledgements.

This project was supported by the National Natural Science Foundation of China
(No. 11665013).

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