# Coupled three-mode squeezed vacuum: Gaussian steering and Remote generation
of Wigner negativity

Zi-wei Zhan, Bo Lan, Jian Wang and Xue-xiang Xu† College of Physics and
Communication Electronics, Jiangxi Normal University, Nanchang 330022, China;
<EMAIL_ADDRESS>
###### Abstract

Coupled three-mode squeezed vacuum (C3MSV), as a typical (and pure) three-mode
Gaussian state, will become an useful entangled resource in future quantum
protocols because of its potential attributes. In this paper, we analyze all
bipartite Gaussian steerings (including no steering, one-way steering and two-
way steering) present in different modes of the C3MSV. Through the steerings
shared in the C3MSV, we propose schemes of remotely generating Wigner
negativity by performing appropriate photon subtraction(s) in the local
position.

Keywords: Gaussian steering; Wigner negativity

##  I Introduction

In many previous quantum protocols, Einstein-Podolshy-Rosen (EPR) state is
perhaps the most commonly used entangled resource1 . Some other entangled
resources, such as N00N state2 ; 3  and Greenberger-Horne-Zeilinger (GHZ)
state4 , have also been used in other different situations. With the
development and requirements of technology, more and more entangled resources
are introduced and used in various protocols5 ; 6 ; 7 ; 8 ; 9 . In current
technology, there is a tread for researchers to use multi-mode or multi-
partite resources. Recently, Zhang and Glasser10  introduced the coupled
three-mode squeezed vacuum (C3MSV), which arises our research interest.
Intuitively, we think that the C3MSV will become an useful entangled resource
in future quantum protocols. Except those properties such as squeezing and
entanglement considered by Zhang and Glasser, we will further study other
properties for the C3MSV in this paper.

Quantum correlations, which is established as a powerful resource in two-
party, three-party or even more-party systems, have been intensively
investigated in recent years and can manifest in different forms, such as
entanglement, steering and Bell nonlocality11 . Entanglement is a striking
feature of describing the nonbiseparability of states for two or more
parties12 . Bell nonlocality offers a vast research landscape with relevance
for fundamental13  and quantum technological applications14 ; 15 . Eistein-
Podolsky-Rosen (EPR) steering is intermediate between entanglement and Bell
nonlocality16 ; 17 ; 18 ; 19 ; 20 . Generally, steerable states are a strict
subset of the entangled states and a strict superset of states that exhibit
Bell nonlocality21 ; 22 . All these correlations can be used as resources for
quantum enhanced tasks23 ; 24 ; 25 ; 26 . On the other hand, Wigner negativity
(WN)27  is arguably one of the most striking non-classical features of quantum
states28 . Beyond its fundamental relevance29 ; 30 , it is also a necessary
resource for quantum speedup with continuous variables. For example, WN has
been seen as a necessary ingredient in CV quantum computation and simulation
to outperform classical devices31 ; 32 .

As two important signatures of nonclassicality, quantum correlations can be
intertwined with Wigner negativity in the conditional generation of non-
Gaussian states33 ; 34 . Walschaers et al. developed a general formalism to
prepare Wigner-negative states through EPR steering35 ; 36 ; 37 . Recently,
Xiang et al. proposed schemes for remote generation of Wigner negativity
through EPR steering in multipartite scenario38 , where they used a pure
three-mode Gaussian state (realized by a feasible linear optical network) as
resource. Following that work, we shall use the C3MSV as entangled resource to
prepare Wigner negative states in this work.

The rest of the paper is structured as follows: In section 2, we make a brief
introduction of the coupled three-mode squeezing operator (C3MSO) and the
coupled three-mode squeezed vacuum (C3MSV). In Section 3, we investigate the
bipartite Gaussian steerings in the C3MSV. In section 4, we propose schemes to
remotely generate Wigner negativity based on the steering in the C3MSV.
Conclusions are summarized in the last section.

##  II Coupled three-mode squeezed vacuum

In nonlinear and quantum optics, there are a number of nonlinear optical
processes to realize various squeezing or multipartite entanglement39 ; 40 .
In this section, we make a brief introduction of the C3MSO and the C3MSV10 .
The C3MSO $S_{3}(\xi_{1},\xi_{2})$ is expressed as the following compact form

$S_{3}=e^{\xi_{1}^{\ast}a_{1}a_{2}+\xi_{2}^{\ast}a_{2}a_{3}-\xi_{1}a_{1}^{\dagger}a_{2}^{\dagger}-\xi_{2}a_{2}^{\dagger}a_{3}^{\dagger}},$
(1)

where $\xi_{j}=r_{j}e^{i\theta_{j}}$ ($j=1,2$) are the two complex squeezing
parameters, with respective magnitude $r_{j}$ and phase $\theta_{j}$. It is
obvious to see $S_{3}^{-1}=S_{3}^{{\dagger}}$. For the convenience purpose, we
reset ($r_{1}$, $r_{2}$) as ($r$, $\phi$), satisfying
$r=\sqrt{r_{1}^{2}+r_{2}^{2}}$, and $\cos\phi=r_{1}/r$, $\sin\phi=r_{2}/r$
with $\phi\in[0,\pi/2]$ (see Fig.1(a)).

As shown in Fig.1(b), we further operate $S_{3}$ on
$\left|0\right\rangle\left|0\right\rangle\left|0\right\rangle$ and obtain the
C3MSV

$\left|\psi\right\rangle\equiv
S_{3}\left|000\right\rangle=\frac{1}{c}e^{-\frac{\epsilon_{1}}{c}a_{1}^{{\dagger}}a_{2}^{{\dagger}}-\frac{\epsilon_{2}}{c}a_{2}^{{\dagger}}a_{3}^{{\dagger}}}\left|000\right\rangle,$
(2)

whose density operator is
$\rho_{123}=\left|\psi\right\rangle\left\langle\psi\right|$. Here, we set
$c=\cosh r$, $s=\sinh r$, $\epsilon_{1}=se^{i\theta_{1}}\cos\phi$, and
$\epsilon_{2}=se^{i\theta_{2}}\sin\phi$. In particular, if $\xi_{2}=0,$ then
$\left|\psi\right\rangle=S_{2}(\xi_{1})\left|00\right\rangle_{12}\otimes\left|0\right\rangle_{3}$;
if $\xi_{1}=0,$ then $\left|\psi\right\rangle=\left|0\right\rangle_{1}\otimes
S_{2}(\xi_{2})\left|00\right\rangle_{23}$, with
$S_{2}(\xi_{1})=e^{\xi_{1}^{\ast}a_{1}a_{2}-\xi_{1}a_{1}^{\dagger}a_{2}^{\dagger}}$
and
$S_{2}(\xi_{2})=e^{\xi_{2}^{\ast}a_{2}a_{3}-\xi_{2}a_{2}^{\dagger}a_{3}^{\dagger}}$.
Moreover, if $\xi_{1}=\xi_{2}$ (i.e. $\phi=\pi/4$), the C3MSV is a bisymmetric
state, whose mode-1 and mode-3 are symmetrical with mode-2.

Using Eq.(A.4), we easily obtain $\bar{n}_{1}=s^{2}\cos^{2}\phi$,
$\bar{n}_{2}=s^{2}$, $\bar{n}_{3}=s^{2}\sin^{2}\phi$ and $\bar{n}_{T}=2s^{2}$,
i.e., the mean photon numbers (MPNs) for mode-1, mode-2, mode-3 and total
modes, respectively (see Fig.1(c)). By the way, we often replace $r$ by
$\bar{n}_{T}$ (using $r=$arcsinh$\sqrt{\bar{n}_{T}/2}$) and set
$\theta_{1}=\theta_{2}=0$ in our numerical works.

Figure 1: (a) ($r_{1}$, $r_{2}$) are reset to ($r$, $\phi$); (b) Conceptual
generating scheme of the C3MSV; (c) MPNs $\bar{n}_{1}$, $\bar{n}_{2}$,
$\bar{n}_{3}$ and $\bar{n}_{T}$ versus $\phi$ (setting $\bar{n}_{T}=2$).

##  III Gaussian steering in the C3MSV

The C3MSV is a pure three-mode entangled Gaussian state, which can be seen
from its Wigner function provided in Eq.(B.0). In this section, we focus on
analyzing the distributions of bipartite Gaussian steerings in the C3MSV.

###  III.1 Covariance Matrix of the C3MSV

The covariance matrix (CM)41 ; 42 ; 43 ; 44 ; 45  of the C3MSV is expressed as

$V=\left(\begin{array}[]{ccc}(1+2\bar{n}_{1})\text{ }1_{2}&-2sc\cos\phi\text{
}\Sigma_{\theta_{1}}&s^{2}\sin 2\phi\text{ }R_{\theta_{2}-\theta_{1}}\\\
-2sc\cos\phi\text{ }\Sigma_{\theta_{1}}&(1+2\bar{n}_{2})\text{
}1_{2}&-2sc\sin\phi\text{ }\Sigma_{\theta_{2}}\\\ s^{2}\sin 2\phi\text{
}\tilde{R}_{\theta_{2}-\theta_{1}}&-2sc\sin\phi\text{
}\Sigma_{\theta_{2}}&(1+2\bar{n}_{3})\text{ }1_{2}\end{array}\right),$ (3)

with $2\times 2$ identity matrix $1_{2}$ and

$\Sigma_{\theta}=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
\sin\theta&-\cos\theta\end{array}\right),R_{\theta}=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right).$ (4)

The matrix elements of the CM, defined by
$V_{jk}=\left\langle\psi\right|(\hat{X}_{j}\hat{X}_{k}+\hat{X}_{k}\hat{X}_{j})\left|\psi\right\rangle$,
are expressed via the vector
$\hat{X}=(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2},\hat{x}_{3},\hat{p}_{3}$).
For each mode, we define position operator
$\hat{x}_{j}=\frac{1}{\sqrt{2}}(\hat{a}_{j}+\hat{a}_{j}^{{\dagger}})$ and
momentum operator
$\hat{p}_{j}=\frac{1}{i\sqrt{2}}(\hat{a}_{j}-\hat{a}_{j}^{{\dagger}})$,
accompanying with its annihilation and creation operators $\hat{a}_{j}$,
$\hat{a}_{j}^{{\dagger}}$. It is noted that
$\left\langle\hat{x}_{j}\right\rangle=\left\langle\hat{p}_{j}\right\rangle=0$
for each mode of the C3MSV. The CM $V$ in Eq.(3) is a symmetric and positive
semidefinite matrix (with eigenvalues $1$, $1$, $e^{-2r}$, $e^{-2r}$,
$e^{2r}$,$e^{2r}$) and obeys $V+i\Omega^{\oplus 3}\geq 0$ with
$\Omega=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$. Moreover, we
can check $\det V=1$ and prove that the C3MSV is a pure state.

###  III.2 Bipartite Gaussian steering

Gaussian steering is a steering by Gaussian measurements46 ; 47 . Reid derived
monogamy inequalities for the bipartite EPR steering distributed among
different systems48 . Xiang et al. derived the laws for the distribution of
quantum steering among different parties and prove a monogamy relation of
Gaussian steering49 . Here, we make a brief review for Gaussian steering.

The CM of a bipartite Gaussian state can be expressed as

$\sigma_{AB}=\left(\begin{array}[]{cc}V_{A}&V_{AB}\\\
V_{AB}^{T}&V_{B}\end{array}\right),$ (5)

where party $A$ and party $B$ are the bipartite subsystems. Then, we can
quantify how much it is steerable via the following quantity

$\mathcal{G}^{A\rightarrow B}\left(V\right):=\max\\{0,-\sum_{j:\text{
}\bar{\nu}_{j}^{B|A}<1}\ln\bar{\nu}_{j}^{B|A}\\}.$ (6)

where $\\{\bar{\nu}_{j}^{B|A}\\}_{j=1}^{2n_{B}}$ denote the symplectic
eigenvalues ($n_{B}$ is the mode number in subsystem $B$) of the Schur
complement $\sigma_{B|A}=V_{B}-V_{AB}^{T}V_{A}^{-1}V_{AB}$ of $\sigma_{AB}$.
This quantity $\mathcal{G}^{A\rightarrow B}$ is defined as Gaussian
$A\rightarrow B$ steerability, which is a monotone under Gaussian local
operations and classical communication. Moreover, the larger
$\mathcal{G}^{A\rightarrow B}$ is, the stronger Gaussian steerability is50 ;
51 .

Figure 2: Bipartite assignments in the C3MSV, where A is the steering party
and B is the steered party.

In what follows, we take party A and party B from the three modes of the C3MSV
and construct 12 kinds of $\sigma_{AB}$s from $V$ in Eq.(3). The steering
party $A$ and the steered party $B$ are assigned as shown in Fig.2. According
to the rule in Eq.(6), we obtain the following steerings present in the C3MSV.

Case (a)$\Longrightarrow$A(23)-B(1): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}+s^{2}\cos 2\phi)^{-1}<1$,
which leads to

$\mathcal{G}^{23\rightarrow 1}=2\ln(c^{2}+s^{2}\cos 2\phi).$ (7)

Case (b)$\Longrightarrow$A(13)-B(2): In this case, we obtain two eigen
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}+s^{2})^{-1}<1$, which leads to

$\mathcal{G}^{13\rightarrow 2}=2\ln(c^{2}+s^{2}).$ (8)

Case (c)$\Longrightarrow$A(12)-B(3): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}-s^{2}\cos 2\phi)^{-1}<1$,
which leads to

$\mathcal{G}^{12\rightarrow 3}=2\ln(c^{2}-s^{2}\cos 2\phi).$ (9)

Case (d)$\Longrightarrow$A(1)-B(23): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=\left(\varkappa_{1}-\sqrt{\varkappa_{2}}\right)/\varkappa_{0}<1$
and
$\bar{\nu}_{3}^{B|A}=\bar{\nu}_{4}^{B|A}=\left(\varkappa_{1}+\sqrt{\varkappa_{2}}\right)/\varkappa_{0}\geq
1$, which leads to

$\mathcal{G}^{1\rightarrow
23}=2\ln[\varkappa_{0}/\left(\varkappa_{1}-\sqrt{\varkappa_{2}}\right)],$ (10)

with

$\displaystyle\varkappa_{0}$ $\displaystyle=$ $\displaystyle
4+8s^{2}\cos^{2}\phi,$ $\displaystyle\varkappa_{1}$ $\displaystyle=$
$\displaystyle 1+3c^{2}+\left(3-2\cos 2\phi\right)s^{2},$
$\displaystyle\varkappa_{2}$ $\displaystyle=$ $\displaystyle\left(19-12\cos
2\phi\right)c^{2}s^{2}$ (11) $\displaystyle+\left(19-12\cos 2\phi+2\cos
4\phi\right)s^{4}$ $\displaystyle+\left(13-20\cos 2\phi\right)s^{2}.$

Case (e)$\Longrightarrow$A(2)-B(13): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}+s^{2})^{-1}<1$ and
$\bar{\nu}_{3}^{B|A}=\bar{\nu}_{4}^{B|A}=1$, which leads to

$\mathcal{G}^{2\rightarrow 13}=2\ln(c^{2}+s^{2}).$ (12)

Case (f)$\Longrightarrow$A(3)-B(12): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=\left(\iota_{1}-\sqrt{\iota_{2}}\right)/\iota_{0}<1$
and
$\bar{\nu}_{3}^{B|A}=\bar{\nu}_{4}^{B|A}=\left(\iota_{1}+\sqrt{\iota_{2}}\right)/\iota_{0}\geq
1$, which leads to

$\mathcal{G}^{3\rightarrow
12}=2\ln[\iota_{0}/\left(\iota_{1}-\sqrt{\iota_{2}}\right)],$ (13)

with

$\displaystyle\iota_{0}$ $\displaystyle=$ $\displaystyle
4+8s^{2}\sin^{2}\phi,$ $\displaystyle\iota_{1}$ $\displaystyle=$
$\displaystyle 1+3c^{2}+(3+2\cos 2\phi)s^{2},$ $\displaystyle\iota_{2}$
$\displaystyle=$ $\displaystyle\left(19+12\cos 2\phi\right)c^{2}s^{2}+$ (14)
$\displaystyle\left(19+12\cos 2\phi+2\cos 4\phi\right)s^{4}$
$\displaystyle+\left(13+20\cos 2\phi\right)s^{2}.$

Case (g)$\Longrightarrow$A(2)-B(1): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}-s^{2}\cos
2\phi)/(c^{2}+s^{2})<1$, which leads to

$\mathcal{G}^{2\rightarrow 1}=2\ln[(c^{2}+s^{2})/(c^{2}-s^{2}\cos 2\phi)].$
(15)

Case (h)$\Longrightarrow$A(1)-B(3): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}+s^{2})/(1+2s^{2}\cos^{2}\phi)\geq
1$,, which leads to

$\mathcal{G}^{1\rightarrow 3}=0.$ (16)

Case (i)$\Longrightarrow$A(2)-B(3): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}+s^{2}\cos
2\phi)/(c^{2}+s^{2})<1$, which leads to

$\mathcal{G}^{2\rightarrow 3}=2\ln[(c^{2}+s^{2})/(c^{2}+s^{2}\cos 2\phi)].$
(17)

Case (j)$\Longrightarrow$A(1)-B(2): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}-s^{2}\cos
2\phi)/(1+2s^{2}\cos^{2}\phi)\geq 1$, which leads to

$\mathcal{G}^{1\rightarrow 2}=0.$ (18)

Case (k)$\Longrightarrow$A(3)-B(1): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}+s^{2})/(1+2s^{2}\sin^{2}\phi)\geq
1$, which leads to

$\mathcal{G}^{3\rightarrow 1}=0.$ (19)

Case (l)$\Longrightarrow$A(3)-B(2): In this case, we have
$\bar{\nu}_{1}^{B|A}=\bar{\nu}_{2}^{B|A}=(c^{2}+s^{2}\cos
2\phi)/(1+2s^{2}\sin^{2}\phi)\geq 1$, which leads to

$\mathcal{G}^{3\rightarrow 2}=0.$ (20)

It is necessary to notice that all above steerings are independent of phases
($\theta_{1}$, $\theta_{2}$). In Fig.3, we draw the contour plots of
$\mathcal{G}^{12\rightarrow 3}$, $\mathcal{G}^{1\rightarrow 23}$,
$\mathcal{G}^{23\rightarrow 1}$, $\mathcal{G}^{3\rightarrow 12}$,
$\mathcal{G}^{2\rightarrow 1}$ and $\mathcal{G}^{2\rightarrow 3}$ in the
($\bar{n}_{T}$, $\phi$)-space. In Fig.4, we plot $\mathcal{G}^{A\rightarrow
B}$s versus $\phi$ (with $\bar{n}_{T}=3$) and $\mathcal{G}^{A\rightarrow B}$s
versus $\bar{n}_{T}$ (with $\phi=\pi/8$). Among them,
$\mathcal{G}^{13\rightarrow 2}=\mathcal{G}^{2\rightarrow 13}$ are functions of
$\bar{n}_{T}$ and are not independent of $\phi$. Moreover, we know that
$\mathcal{G}^{1\rightarrow 2}=\mathcal{G}^{1\rightarrow
3}=\mathcal{G}^{3\rightarrow 1}=\mathcal{G}^{3\rightarrow 2}=0$, but
$\mathcal{G}^{2\rightarrow 1}>0$ (except $\phi=\pi/2$),
$\mathcal{G}^{2\rightarrow 3}>0$ (except $\phi=0$) and
$\mathcal{G}^{2\rightarrow 1}=\mathcal{G}^{2\rightarrow 3}$ for $\phi=\pi/4$.
These results tell us that three types of steerings, i.e., no steering ($A$
cannot steer $B$ and $B$ cannot steer $A$), one-way steering ($A$ can steer
$B$ while $B$ cannot steer $A$), or two-way (symmetrical or asymmetrical)
steering ($A$ can steer $B$ and $B$ can steer $A$), are presented in the
C3MSV. Main results are summarized as:

(i) There is no steering between mode 1 and mode 3 because of
$\mathcal{G}^{1\rightarrow 3}=0$ and $\mathcal{G}^{3\rightarrow 1}=0$, see
Eq.(16) and Eq.(19);

(ii) There is one-way steering between mode 1 and mode 2 because of
$\mathcal{G}^{2\rightarrow 1}>0$ and $\mathcal{G}^{1\rightarrow 2}=0$, see
Eq.(15) and Eq.(18);

(iii) There is one-way steering between mode 2 and mode 3 because of
$\mathcal{G}^{2\rightarrow 3}>0$ and $\mathcal{G}^{3\rightarrow 2}=0$, see
Eq.(17) and Eq.(20);

(iv) There is two-way asymmetrical steering between mode 1 and modes 23
because of $\mathcal{G}^{23\rightarrow 1}>0$ and $\mathcal{G}^{1\rightarrow
23}>0$ but $\mathcal{G}^{23\rightarrow 1}\neq\mathcal{G}^{1\rightarrow 23}$,
see Eq.(7) and Eq.(10);

(v) There is two-way symmetrical steering between mode 2 and modes 13 because
of $\mathcal{G}^{13\rightarrow 2}=$ $\mathcal{G}^{2\rightarrow 13}>0$, see
Eq.(8) and Eq.(12);

(vi) There is two-way asymmetrical steering between mode 3 and modes 12
because of $\mathcal{G}^{12\rightarrow 3}>0$ and $\mathcal{G}^{3\rightarrow
12}>0$ but $\mathcal{G}^{12\rightarrow 3}\neq\mathcal{G}^{3\rightarrow 12}$,
see Eq.(9) and Eq.(13).

Just like what He et al. said in their work20 , our results also show that
each mode can be steered by one or both of the other two in the C3MSV.
Moreover, we find that (a) $\mathcal{G}^{2\rightarrow 1}>0$, but
$\mathcal{G}^{3\rightarrow 1}=0$ and (b) $\mathcal{G}^{2\rightarrow 3}>0$, but
$\mathcal{G}^{1\rightarrow 3}=0$. This result holds the character that two
parties cannot steer the same system48 .

Figure 3: (a) $\mathcal{G}^{12\rightarrow 3}$, (b) $\mathcal{G}^{23\rightarrow
1}$, (c) $\mathcal{G}^{1\rightarrow 23}$, (d) $\mathcal{G}^{3\rightarrow 12}$,
(e) $\mathcal{G}^{2\rightarrow 1}$ and (f) $\mathcal{G}^{2\rightarrow 3}$ as
functions of $\bar{n}_{T}$ and $\phi$.

Figure 4: (a) $\mathcal{G}^{A\rightarrow B}$ versus $\phi$ (setting
$\bar{n}_{T}=3$); (b) $\mathcal{G}^{A\rightarrow B}$ versus $\bar{n}_{T}$
(setting $\phi=\pi/8$).

###  III.3 Monogamy relations

In 2017, Xiang et al. defined the concept of the residual Gaussian steering
(RGS)49 . Here, we shall use the RGS to quantify the genuine tripartite
steering for the C3MSV. Using all above expressions from Eq.(7) to Eq.(20), we
check that the monogamy relations

$\displaystyle\mathcal{G}^{\left(23\right)\rightarrow
1}-\mathcal{G}^{2\rightarrow 1}-\mathcal{G}^{3\rightarrow 1}$
$\displaystyle\geq$ $\displaystyle 0,$
$\displaystyle\mathcal{G}^{\left(31\right)\rightarrow
2}-\mathcal{G}^{3\rightarrow 2}-\mathcal{G}^{1\rightarrow 2}$
$\displaystyle\geq$ $\displaystyle 0,$
$\displaystyle\mathcal{G}^{\left(12\right)\rightarrow
3}-\mathcal{G}^{1\rightarrow 3}-\mathcal{G}^{2\rightarrow 3}$
$\displaystyle\geq$ $\displaystyle 0,$ (21)

and

$\displaystyle\mathcal{G}^{1\rightarrow\left(23\right)}-\mathcal{G}^{1\rightarrow
2}-\mathcal{G}^{1\rightarrow 3}$ $\displaystyle\geq$ $\displaystyle 0,$
$\displaystyle\mathcal{G}^{2\rightarrow\left(31\right)}-\mathcal{G}^{2\rightarrow
3}-\mathcal{G}^{2\rightarrow 1}$ $\displaystyle\geq$ $\displaystyle 0,$
$\displaystyle\mathcal{G}^{3\rightarrow\left(12\right)}-\mathcal{G}^{3\rightarrow
1}-\mathcal{G}^{3\rightarrow 2}$ $\displaystyle\geq$ $\displaystyle 0,$ (22)

hold for the C3MSV. Further, we consider the RGS

$\displaystyle\mathcal{G}^{1:2:3}$ $\displaystyle=$
$\displaystyle\min_{\left\langle
i,j,k\right\rangle}\\{\mathcal{G}^{\left(jk\right)\rightarrow
i}-\mathcal{G}^{j\rightarrow i}-\mathcal{G}^{k\rightarrow i}\\}$ (23)
$\displaystyle=$ $\displaystyle\min_{\left\langle
i,j,k\right\rangle}\\{\mathcal{G}^{i\rightarrow\left(jk\right)}-\mathcal{G}^{i\rightarrow
j}-\mathcal{G}^{i\rightarrow k}\\},$

for the C3MSV, where $\left\langle i,j,k\right\rangle$ denotes any cycle
permutation of $1$, $2$, $3$. In Fig.5(a) we plot the RGS as a function of
$\bar{n}_{T}$ and $\phi$. From which, we see that the RGS is maximized on
bisymmetric C3MSV with $\phi=\pi/4$, i.e., $r_{1}=r_{2}$. In this case, the
genuine tripartite $\mathcal{G}^{1:2:3}$ reduces to the collective steering
$\mathcal{G}^{13\rightarrow 2}=\mathcal{G}^{2\rightarrow
13}=2\ln(c^{2}+s^{2})$. Figure 5(b) presents the RGS as a function of $\phi$
with different $\bar{n}_{T}$, which are the sections of Fig.5(a). Indeed, the
RGS acts as an indicator of collective steering-type correlations.

Figure 5: RGS $\mathcal{G}^{1:2:3}$ for the C3MSV with CM $V$ (a) in the
($\bar{n}_{T}$, $\phi$) space; (b) versus $\phi$ with different $\bar{n}_{T}$.

##  IV Remotely generated Wigner negativity

As Walschaers recently pointed out, when party $A$ and party $B$ share a
Gaussian state, party $B$ can perform some measurement on itself to create
Wigner negativity on party $A$, if and only if there is a Gaussian steering
from party $A$ to party $B$33 . Following ways in Xiang’s work38  and
Walschaers’ work35 , we investigate the remote creation and distribution of
Wigner negativity (WN) in the tripartite C3MSV. Based on the C3MSV, we remain
the steered party $B$ in the local station and sent the steering party $A$ to
the remote position. After appropriate single-photon subtraction(s) on the
steered party $B$, the steering party $A$ becomes a reduced non-Gaussian state
$\rho_{B_{a}|A}$. In some cases, we can generate Wigner negative states in the
remote position.

Figure 6: Conceptual schemes of remote generated non-Gaussian states
$\rho_{B_{a}|A}$ based on the C3MSV. Note that Cases (a)-(c) are two-mode
states and Cases (d)-(l) are one-mode states. Figure 7: WFs of
$\rho_{B_{a}|A}$s corresponding to Fig.6, with $\bar{n}_{T}=3$, $\phi=\pi/8$,
and $\theta_{1}=\theta_{2}=0$. Some cases have WNs and some cases have no WNs.

For state $\rho_{j}$, we can derive its Wigner function by
$W_{\rho_{j}}\left(\gamma_{j}\right)=\mathrm{Tr}(\hat{O}_{w_{j}}\rho_{j})$,
with
$\hat{O}_{w_{j}}=\frac{2}{\pi}:e^{-2(\hat{a}_{j}^{{\dagger}}-\gamma_{j}^{\ast})(\hat{a}_{j}-\gamma_{j})}:$
($:\cdots:$ denotes the normal ordering) and
$\gamma_{j}=\left(x_{j}+iy_{j}\right)/\sqrt{2}$. Furthermore, we can quantify
the WN of $\rho_{B_{a}|A}$ as

$\mathcal{N}\equiv\int\left|W(\gamma)\right|d^{2n_{A}}\gamma-1,$ (24)

with $\gamma\in\mathbb{R}^{2n_{A}}$, where $n_{A}$ is the mode number
considered in party $A$. As shown schematically in Fig.6, we propose protocols
of generating 18 kinds of $\rho_{B_{a}|A}$s, whose analytical WFs are given in
Appendix B. As examples, we plot WFs for $\rho_{B_{a}|A}$s with
$\bar{n}_{T}=3$ and $\phi=\pi/8$ in Fig.7, where only several WFs exhibits the
WNs.

Indeed, the amount of WN cannot be freely distributed among different modes.
It can be influenced by the considered protocols and the interaction
parameters. In order to explain the characters, we plot some WN $\mathcal{N}$
versus $\phi$ by fixing $\bar{n}_{T}=3$ in Fig.8 and versus $\bar{n}_{T}$ by
fixing $\phi=\pi/8$ in Fig.9. The details are explained as follows.

Figure 8: WNs versus $\phi$ with fixed $\bar{n}_{T}=3$, for some
$\rho_{B_{a}|A}$s in cases (a) $A(23)B(1)$, (b) $A(13)B(2)$, (c) $A(12)B(3)$,
(d) $A(2)B(13)$. Figure 9: WNs versus $\bar{n}_{T}$ with fixed $\phi=\pi/8$,
for some $\rho_{B_{a}|A}$s in cases (a) $A(23)B(1)$, (b) $A(13)B(2)$, (c)
$A(12)B(3)$, (d) $A(2)B(13)$.

Case $A(23)B(1)$: In this case, the steering party $A$ includes mode-2 and
mode-3 and the steered party $B$ includes mode-1. Performing appropriate
photon subtraction(s) in the local position, we can remotely generate the
following states with their respective WN

$\displaystyle\rho_{1_{a}|23}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1}\left(\rho_{1_{a}23}\right)\rightarrow\mathcal{N}_{1_{a}|23},$
$\displaystyle\rho_{1_{a}|2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,3}\left(\rho_{1_{a}23}\right)\rightarrow\mathcal{N}_{1_{a}|2},$
$\displaystyle\rho_{1_{a}|3}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,2}\left(\rho_{1_{a}23}\right)\rightarrow\mathcal{N}_{1_{a}|3},$
(25)

where
$\rho_{1_{a}23}=\left|\epsilon_{1}\right|_{1}^{-2}\hat{a}_{1}\rho_{123}\hat{a}_{1}^{{\dagger}}$.

We plot $\mathcal{N}_{1_{a}|23}$, $\mathcal{N}_{1_{a}|2}$, and
$\mathcal{N}_{1_{a}|3}$ as functions of $\phi$ in Fig.8(a) and as functions of
$\bar{n}_{T}$ in Fig.9(a). From these figures, we find that WNs are generated
remotely in the group (23), mode-2, mode-3, respectively, after a single-
photon subtraction on mode-1. Moreover, we see that (1)
$\mathcal{N}_{1_{a}|23}$ is a monotonically increasing function of $\phi$ from
$0.04682$ at $\phi=0$ to $0.42614$ at $\phi=\pi/2$; (2)
$\mathcal{N}_{1_{a}|2}\ $ remains as $0.04682$ for any $\phi$; (3)
$\mathcal{N}_{1_{a}|3}\ $ remains as $0$ for any $\phi$; (4)
$\mathcal{N}_{1_{a}|23}\geq\mathcal{N}_{1_{a}|2}+\mathcal{N}_{1_{a}|3}$; (5)
As $\bar{n}_{T}$ increasing, all these WNs will limit to $0$.

Case $A(12)B(3)$: In this case, the steering party $A$ include mode-1 and
mode-2 and the steered party $B$ include mode-3. Performing appropriate photon
subtraction(s) in the local position, we can remotely generate the following
states with their respective WN

$\displaystyle\rho_{3_{a}|12}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{3}\left(\rho_{123_{a}}\right)\rightarrow\mathcal{N}_{3_{a}|12},$
$\displaystyle\rho_{3_{a}|1}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{2,3}\left(\rho_{123_{a}}\right)\rightarrow\mathcal{N}_{3_{a}|1},$
$\displaystyle\rho_{3_{a}|2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,3}\left(\rho_{123_{a}}\right)\rightarrow\mathcal{N}_{3_{a}|2},$
(26)

where
$\rho_{123_{a}}=\left|\epsilon_{2}\right|^{-2}\hat{a}_{3}\rho_{123}\hat{a}_{3}^{{\dagger}}$.

We plot $\mathcal{N}_{3_{a}|12}$, $\mathcal{N}_{3_{a}|1}$, and
$\mathcal{N}_{3_{a}|2}$ as functions of $\phi$ in Fig.8(b) and as functions of
$\bar{n}_{T}$ in Fig.9(b). From these figures, we find that WNs are generated
remotely in the group (12), mode-1, mode-2, respectively, after a single-
photon subtraction on mode-3. Moreover, we see that (1)
$\mathcal{N}_{3_{a}|12}$ is a monotonically decreasing function of $\phi$ from
$0.42614$ at $\phi=0$ to $0.04682$ at $\phi=\pi/2$; (2)
$\mathcal{N}_{3_{a}|2}\ $ remains as $0.04682$ for any $\phi$; (3)
$\mathcal{N}_{3_{a}|1}\ $ remains as $0$ for any $\phi$; (4)
$\mathcal{N}_{3_{a}|12}\geq\mathcal{N}_{3_{a}|1}+\mathcal{N}_{3_{a}|2}$; (5)
As $\bar{n}_{T}$ increasing, all these WNs will limit to $0$.

Case $A(13)B(2)$: In this case, the steering party $A$ includes mode-1 and
mode-3 and the steered party $B$ includes mode-2. Performing appropriate
photon subtraction(s) in the local position, we can remotely generate the
following states with their respective WN

$\displaystyle\rho_{2_{a}|13}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{2}\left(\rho_{12_{a}3}\right)\rightarrow\mathcal{N}_{2_{a}|13},$
$\displaystyle\rho_{2_{a}|1}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{2,3}\left(\rho_{12_{a}3}\right)\rightarrow\mathcal{N}_{2_{a}|1},$
$\displaystyle\rho_{2_{a}|3}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,2}\left(\rho_{12_{a}3}\right)\rightarrow\mathcal{N}_{2_{a}|3},$
(27)

where $\rho_{12_{a}3}=s^{-2}\hat{a}_{2}\rho_{123}\hat{a}_{2}^{{\dagger}}$.

We plot $\mathcal{N}_{2_{a}|13}$, $\mathcal{N}_{2_{a}|1}$, and
$\mathcal{N}_{2_{a}|3}$ as functions of $\phi$ in Fig.8(c) and as functions of
$\bar{n}_{T}$ in Fig.9(c). From these figures, we find that WNs are generated
remotely in the group (13), mode-1, mode-3, respectively, after a single-
photon subtraction on mode-2. Moreover, we see that (1)
$\mathcal{N}_{2_{a}|13}$ remains as $0.4683$ for any $\phi$; (2)
$\mathcal{N}_{2_{a}|1}$ decreases from $0.4683$ to $0$ in
$\left[0,\pi/4\right]$ and remains as $0$ in $\left[\pi/4,\pi/2\right]$; (3)
$\mathcal{N}_{2_{a}|3}$ remains as $0$ in $\left[0,\pi/4\right]$ and increases
from $0$ to $0.4683$ in $\left[\pi/4,\pi/2\right]$; (4)
$\mathcal{N}_{2_{a}|13}\geq\mathcal{N}_{2_{a}|1}+\mathcal{N}_{2_{a}|3}$; (5)
As $\bar{n}_{T}$ increasing, all these WNs will limit to $0$.

Case $A(2)B(13)$: In this case, the steering party $A$ include mode-2 and the
steered party $B$ include mode-1 and mode-3. Performing appropriate photon
subtraction(s) in the local position, we can remotely generate the following
states with their respective WN

$\displaystyle\rho_{1_{a}3_{a}|2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,3}\left(\rho_{1_{a}23_{a}}\right)\rightarrow\mathcal{N}_{1_{a}3_{a}|2},$
$\displaystyle\rho_{1_{a}3|2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,3}\left(\rho_{1_{a}23}\right)\rightarrow\mathcal{N}_{1_{a}3|2},$
$\displaystyle\rho_{13_{a}|2}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,3}\left(\rho_{123_{a}}\right)\rightarrow\mathcal{N}_{13_{a}|2},$
(28)

where
$\rho_{1_{a}23_{a}}=\frac{1}{2}\left|\epsilon_{1}\epsilon_{2}\right|^{-2}\hat{a}_{1}\hat{a}_{3}\rho_{123}\hat{a}_{1}^{{\dagger}}\hat{a}_{3}^{{\dagger}}$.

We plot $\mathcal{N}_{1_{a}3_{a}|2}$, $\mathcal{N}_{1_{a}3|2}$, and
$\mathcal{N}_{13_{a}|2}$ as functions of $\phi$ in Fig.8(d) and as functions
of $\bar{n}_{T}$ in Fig.9(d). From these figures, we find that WNs are
generated remotely in mode-2, after single-photon subtractions on each of the
group (13), or a single-photon subtraction on mode-1 or mode-3, respectively.
Here, we see that (1) $\mathcal{N}_{1_{a}3_{a}|2}$ remins as $0.0318528$, (2)
$\mathcal{N}_{1_{a}3|2}$ and $\mathcal{N}_{13_{a}|2}\ $remain as $0.04683$;
(3)
$\mathcal{N}_{1_{a}3_{a}|2}<\mathcal{N}_{1_{a}3|2}+\mathcal{N}_{13_{a}|2}$;
(4) As $\bar{n}_{T}$ increasing, all these WNs will limit to $0$. However,
although $\mathcal{G}^{2\rightarrow 1}>0$, $\mathcal{G}^{2\rightarrow 3}>0$,
and $\mathcal{G}^{2\rightarrow\left(31\right)}>0$, we cannot achieve more
significant increase of the WNs in mode-2, after performing a single-photon
subtraction on each of mode-1 and mode-3.

Case $A(1)B(23)$: In this case, the steering party $A$ include mode-1 and the
steered party $B$ include mode-2 and mode-3. Performing appropriate photon
subtraction(s) in the local position, we can remotely generate the following
states with their respective WN

$\displaystyle\rho_{2_{a}3_{a}|1}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{2,3}\left(\rho_{12_{a}3_{a}}\right)\rightarrow\mathcal{N}_{2_{a}3_{a}|1}\equiv
0,$ $\displaystyle\rho_{2_{a}3|1}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{2,3}\left(\rho_{12_{a}3}\right)\rightarrow\mathcal{N}_{2_{a}3|1}\equiv
0,$ $\displaystyle\rho_{23_{a}|1}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{2,3}\left(\rho_{123_{a}}\right)\rightarrow\mathcal{N}_{23_{a}|1}\equiv
0,$ (29)

where
$\rho_{12_{a}3_{a}}=\left(c^{2}+s^{2}\right)^{-1}\left|\epsilon_{2}\right|^{-2}\hat{a}_{2}\hat{a}_{3}\rho_{123}\hat{a}_{2}^{{\dagger}}\hat{a}_{3}^{{\dagger}}$.
For any $\bar{n}_{T}$ and $\phi$, we see
$\mathcal{N}_{2_{a}3_{a}|1}=\mathcal{N}_{2_{a}3|1}=\mathcal{N}_{23_{a}|1}\equiv
0$. That is to say, no WNs are generated remotely in mode-1, after single-
photon subtractions on each of the group (23), or a single-photon subtraction
on mode-2, mode-3. Surprisingly, $\mathcal{N}_{2_{a}3_{a}|1}=0$ although
$\mathcal{G}^{1\rightarrow 23}>0$.

Case $A(3)B(12)$: In this case, the steering party $A$ include mode-3 and the
steered party $B$ include mode-1 and mode-2. Performing appropriate photon
subtraction(s) in the local position, we can remotely generate the following
states with their respective WN

$\displaystyle\rho_{1_{a}2_{a}|3}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,2}\left(\rho_{1_{a}2_{a}3}\right)\rightarrow\mathcal{N}_{1_{a}2_{a}|3}\equiv
0,$ $\displaystyle\rho_{1_{a}2|3}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,2}\left(\rho_{1_{a}23}\right)\rightarrow\mathcal{N}_{1_{a}2|3}\equiv
0,$ $\displaystyle\rho_{12_{a}|3}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{1,2}\left(\rho_{12_{a}3}\right)\rightarrow\mathcal{N}_{12_{a}|3}\equiv
0,$ (30)

where
$\rho_{1_{a}2_{a}3}=\left(c^{2}+s^{2}\right)^{-1}\left|\epsilon_{1}\right|^{-2}\hat{a}_{1}\hat{a}_{2}\rho_{123}\hat{a}_{1}^{{\dagger}}\hat{a}_{2}^{{\dagger}}$.
For any $\bar{n}_{T}$ and $\phi$, we see
$\mathcal{N}_{1_{a}2_{a}|3}=\mathcal{N}_{1_{a}2|3}=\mathcal{N}_{12_{a}|3}\equiv
0$. That is to say, no WNs are generated remotely in mode-3, after single-
photon subtractions on each of the group (12), or a single-photon subtraction
on mode-1, mode-2. Surprisingly, $\mathcal{N}_{1_{a}2_{a}|3}=0$ although
$\mathcal{G}^{3\rightarrow 12}>0$.

##  V Conclusion

In this paper, we studied the steering property for the C3MSV and used the
C3MSV as resource to generate Wigner-negative state. By taking different
bipartite assignment in the C3MSV, we investigated all bipartite Gaussian
steerings present in the C3MSV. These steerings include no seering, one-way
steering and two-way steering. Moreover, the steerability can be adjusted by
the interaction parameters. Using the C3MSV as the resource, we proposed
schemes to remotely generate Wigner negative states. We analyzed and compared
the distributions the Gaussian steering and the WNs over different modes.

Normally, if $\mathcal{G}^{A\rightarrow B}>0$, then $\mathcal{N}>0$; if
$\mathcal{G}^{A\rightarrow B}=0$, then $\mathcal{N}=0$. However, quantum
correlations are not always a necessary requirement for the conditional
generation of Wigner negativty33 . For example, although
$\mathcal{G}^{1\rightarrow 23}>0$ and $\mathcal{G}^{3\rightarrow 12}>0$,
$\rho_{2_{a}3_{a}|1}$ and $\rho_{1_{a}2_{a}|3}$ cannot exhibit WN. Moreover,
it is not always true that stronger steerability in the C3MSV induces more
WNs.

We believe that the C3MSV will become an useful entangled resource in future
quantum protocols. Its specific properties (including squeezing, entanglement
and steering) has laid a good foundation for its applications in quantum
technologies.

###### Acknowledgements.

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

## Appendix A: The C3MSO and the C3MSV

In this appendix, we give the transformation relation and the normal ordering
form for the C3MSO. In addition, we give the general expression to calculate
the expectation values we want for the C3MSV.

About the C3MSO: Using the formula of Bogoliubov transformation, we obtain the
following transformation relations

$S_{3}A^{{\dagger}}S_{3}^{{\dagger}}=A^{{\dagger}}P^{\ast}+AL^{\ast},S_{3}AS_{3}^{{\dagger}}=AP+A^{{\dagger}}L,$
(A.1)

where
$A^{{\dagger}}=\left(\hat{a}_{1}^{{\dagger}},\hat{a}_{2}^{{\dagger}},\hat{a}_{3}^{{\dagger}}\right)$,
$A=\left(\hat{a}_{1},\hat{a}_{2},\hat{a}_{3}\right)$, and

$P=\left(\begin{array}[]{ccc}\kappa_{1}&0&\tau\\\ 0&c&0\\\
\tau^{\ast}&0&\kappa_{2}\end{array}\right),L=\left(\begin{array}[]{ccc}0&\epsilon_{1}&0\\\
\epsilon_{1}&0&\epsilon_{2}\\\ 0&\epsilon_{2}&0\end{array}\right).$ (A.2)

Here we set $\kappa_{1}=\sin^{2}\phi+c\cos^{2}\phi$,
$\kappa_{2}=\cos^{2}\phi+c\sin^{2}\phi$, and
$\tau=\frac{1}{2}(c-1)e^{i(\theta_{2}-\theta_{1})}\sin 2\phi$.

According to the rule provided by Fan52 ; 53 ; 54 , we immediately obtain the
normal ordering form of $S_{3}$ as follows

$\displaystyle S_{3}$ $\displaystyle=\frac{1}{\sqrt{\det
P}}e^{-\frac{1}{2}A^{{\dagger}}(LP^{-1})\tilde{A}^{{\dagger}}}$
$\displaystyle:e^{A^{{\dagger}}(\tilde{P}^{-1}-I)\tilde{A}}:e^{\frac{1}{2}A(P^{-1}L^{\ast})\tilde{A}}.$
(A.3)

Of course, we can further use
$:e^{A^{{\dagger}}(\tilde{P}^{-1}-I)\tilde{A}}:=e^{A^{{\dagger}}(\ln\tilde{P}^{-1})\tilde{A}}$
in above expression, where $\tilde{P}$ denotes the transpose of $P$.

About the C3MSV: Here, we give the following general expression

$\displaystyle\left\langle\hat{a}_{1}^{{\dagger}k_{1}}\hat{a}_{2}^{{\dagger}k_{2}}\hat{a}_{3}^{{\dagger}k_{3}}\hat{a}_{1}^{l_{1}}\hat{a}_{2}^{l_{2}}\hat{a}_{3}^{l_{3}}\right\rangle$
$\displaystyle=\partial_{\mu_{1}}^{k_{1}}\partial_{\mu_{2}}^{k_{2}}\partial_{\mu_{3}}^{k_{3}}\partial_{\nu_{1}}^{l_{1}}\partial_{\nu_{2}}^{l_{2}}\partial_{\nu_{3}}^{l_{3}}$
$\displaystyle
e^{\left|\epsilon_{1}\right|^{2}\mu_{1}\nu_{1}+s^{2}\mu_{2}\nu_{2}+\left|\epsilon_{2}\right|^{2}\mu_{3}\nu_{3}+\epsilon_{1}^{\ast}\epsilon_{2}\mu_{1}\nu_{3}+\epsilon_{1}\epsilon_{2}^{\ast}\mu_{3}\nu_{1}}$
$\displaystyle
e^{-c\epsilon_{1}^{\ast}\mu_{1}\mu_{2}-c\epsilon_{1}\nu_{1}\nu_{2}-c\epsilon_{2}^{\ast}\mu_{2}\mu_{3}-c\epsilon_{2}\nu_{2}\nu_{3}}$
$\displaystyle|_{\mu_{1}=\mu_{2}=\mu_{3}=\nu_{1}=\nu_{2}=\nu_{3}=0},$ (A.4)

from which one can study the statistical properties for the C3MSV.

## Appendix B: Wigner functions of remotely generated states

The WF for the C3MSV is

$\displaystyle W_{\rho_{123}}$
$\displaystyle=\frac{8}{\pi^{3}}e^{-2[\left(2\bar{n}_{1}+1\right)\left|\gamma_{1}\right|^{2}+\left(2\bar{n}_{2}+\allowbreak
1\right)\left|\gamma_{2}\right|^{2}+\left(2\bar{n}_{3}+1\right)\left|\gamma_{3}\right|^{2}]}$
$\displaystyle\times
e^{-8[\text{Re}(c\epsilon_{1}^{\ast}\gamma_{1}\gamma_{2}\allowbreak)+\text{Re}(\epsilon_{1}^{\ast}\epsilon_{2}\gamma_{1}\gamma_{3}^{\ast})+\text{Re}(c\epsilon_{2}^{\ast}\allowbreak\gamma_{2}\gamma_{3})]},$
(B.0)

which has the Gaussian form.

The analytical WFs of $\rho_{B_{a}|A}$s are given as follows. For convenience
of writing, we set $\omega_{0}=c^{2}+s^{2}=\cosh 2r$,
$\omega_{1}=c^{2}-s^{2}\cos 2\phi$, $\omega_{2}=c^{2}+s^{2}\cos 2\phi$. If
$\phi=0$, then $\epsilon_{2}=0$, $r=r_{1}$, $\omega_{1}=1$ and
$\omega_{2}=\omega_{0}$; If $\phi=\pi/2$, then $\epsilon_{1}=0$, $r=r_{2}$,
$\omega_{1}=\omega_{0}$ and $\omega_{2}=1$.

(a) WF for $\rho_{1_{a}|23}$

$\displaystyle W_{\rho_{1_{a}|23}}$
$\displaystyle=\frac{4e^{-2[\omega_{1}\left|\gamma_{2}\right|^{2}+\omega_{0}\left|\gamma_{3}\right|^{2}+4\text{Re}(c\epsilon_{2}^{\ast}\gamma_{2}\gamma_{3})]/\omega_{2}}}{\pi^{2}\omega_{2}^{3}}$
$\displaystyle\times(4\left|c\gamma_{2}+\epsilon_{2}\gamma_{3}^{\ast}\right|^{2}-\omega_{2}).$
(B.1)

When $\phi=0$, Eq.(B.1) will be reduced to
$W_{\rho_{1_{a}|23}}=(2/\pi)e^{-2\left|\gamma_{2}\right|^{2}/\omega_{0}}(4c^{2}\omega_{0}^{-3}\left|\gamma_{2}\right|^{2}-\omega_{0}^{-2})\times(2/\pi)e^{-2\left|\gamma_{3}\right|^{2}}$.

(b) WF for $\rho_{2_{a}|13}$

$\displaystyle W_{\rho_{2_{a}|13}}$
$\displaystyle=\frac{4e^{-2[\omega_{1}\left|\gamma_{1}\right|^{2}+\omega_{2}\allowbreak\left|\gamma_{3}\right|^{2}-4\text{Re}(\epsilon_{1}^{\ast}\epsilon_{2}\gamma_{1}\allowbreak\gamma_{3}^{\ast})]/\omega_{0}}}{\pi^{2}s^{2}\omega_{0}^{3}}$
$\displaystyle\times(4c^{2}\left|\epsilon_{1}^{\ast}\gamma_{1}+\epsilon_{2}^{\ast}\gamma_{3}\right|^{2}-\omega_{0}s^{2}\allowbreak).$
(B.2)

When $\phi=0$, Eq.(B.3) will be reduced to
$W_{\rho_{2_{a}|13}}=(2/\pi)e^{-2\left|\gamma_{1}\right|^{2}/\omega_{0}}(4c^{2}\omega_{0}^{-3}\cos^{2}\phi\left|\gamma_{1}\right|^{2}-\omega_{0}^{-2})\times(2/\pi)e^{-2\left|\gamma_{3}\right|^{2}}$;
When $\phi=\pi/2$, Eq.(B.3) will be reduced to
$W_{\rho_{2_{a}|13}}=(2/\pi)e^{-2\left|\gamma_{3}\right|^{2}/\omega_{0}}(4c^{2}\omega_{0}^{-3}\sin^{2}\phi\left|\gamma_{3}\right|^{2}-\omega_{0}^{-2})\times(2/\pi)e^{-2\left|\gamma_{1}\right|^{2}}$.

(c) WF for $\rho_{3_{a}|12}$

$\displaystyle W_{\rho_{3_{a}|12}}$
$\displaystyle=\frac{4e^{-2[\omega_{2}\left|\gamma_{2}\right|^{2}+\omega_{0}\left|\gamma_{1}\right|^{2}+4\text{Re}(c\epsilon_{1}^{\ast}\gamma_{1}\gamma_{2})]/\omega_{1}}}{\pi^{2}\omega_{1}^{3}}$
$\displaystyle\times(4\left|c\gamma_{2}+\epsilon_{1}\gamma_{1}^{\ast}\right|^{2}-\omega_{1}).$
(B.3)

When $\phi=\pi/2$, Eq.(B.2) will be reduced to
$W_{\rho_{3_{a}|12}}=(2/\pi)e^{-2\left|\gamma_{2}\right|^{2}/\omega_{0}}(4c^{2}\omega_{0}^{-3}\left|\gamma_{2}\right|^{2}-\omega_{0}^{-2})\times(2/\pi)e^{-2\left|\gamma_{1}\right|^{2}}$.

(d) WF for $\rho_{2_{a}3_{a}|1}$

$\displaystyle W_{\rho_{2_{a}3_{a}|1}}$
$\displaystyle=\frac{2e^{-2\left|\gamma_{1}\right|^{2}/\omega_{2}}}{\pi\omega_{0}\omega_{2}^{5}}$
$\displaystyle\times[\omega_{2}^{2}\allowbreak\omega_{1}+4\left|\epsilon_{1}\gamma_{1}\right|^{2}(4c^{4}+4c^{2}\left|\epsilon_{1}\gamma_{1}\right|^{2}-\omega_{1}^{2})]$
(B.4)

When $\phi=\pi/2$, Eq.(B.4) will be reduced to
$W_{\rho_{2_{a}3_{a}|1}}=(2/\pi)e^{-2\left|\gamma_{1}\right|^{2}}$.

(e) WF for $\rho_{1_{a}3_{a}|2}$

$W_{\rho_{1_{a}3_{a}|2}}=\frac{2e^{-2\left|\gamma_{2}\right|^{2}/\omega_{0}}}{\pi\omega_{0}^{5}}(\omega_{0}^{2}\allowbreak+8c^{4}\left|\gamma_{2}\right|^{4}-8\omega_{0}c^{2}\left|\gamma_{2}\right|^{2}),$
(B.5)

which is independent of $\phi$.

(f) WF for $\rho_{1_{a}2_{a}|3}$

$\displaystyle W_{\rho_{1_{a}2_{a}|3}}$
$\displaystyle=\frac{2e^{-2\left|\gamma_{3}\right|^{2}/\omega_{1}}}{\pi\omega_{0}\omega_{1}^{5}}\allowbreak$
$\displaystyle\times[\omega_{1}^{2}\allowbreak\omega_{2}+4\left|\epsilon_{2}\gamma_{3}\right|^{2}(4c^{4}\allowbreak+4c^{2}\left|\epsilon_{2}\gamma_{3}\right|^{2}-\omega_{2}^{2})].$
(B.6)

When $\phi=0$, Eq.(B.6) will be reduced to
$W_{\rho_{1_{a}2_{a}|3}}=(2/\pi)e^{-2\left|\gamma_{3}\right|^{2}}$.

(g) WF for $\rho_{1_{a}|2}$ and $\rho_{1_{a}3|2}$

$W_{\rho_{1_{a}|2}}=W_{\rho_{1_{a}3|2}}=\frac{2e^{-2\left|\gamma_{2}\right|^{2}/\omega_{0}}}{\pi\omega_{0}^{3}}(4c^{2}\left|\gamma_{2}\right|^{2}-\omega_{0}),$
(B.7)

which is independent of $\phi$.

(h) WF for $\rho_{3_{a}|1}$ and $\rho_{23_{a}|1}$

$W_{\rho_{3_{a}|1}}=W_{\rho_{23_{a}|1}}=\frac{\allowbreak 2e^{-\allowbreak
2\left|\gamma_{1}\right|^{2}/\omega_{2}}}{\pi\omega_{2}^{3}}(4\left|\epsilon_{1}\gamma_{1}\right|^{2}+\omega_{2}).$
(B.8)

When $\phi=\pi/2$, Eq.(B.8) will be reduced to
$W_{\rho_{3_{a}|1}}=W_{\rho_{23_{a}|1}}=(2/\pi)e^{-2\left|\gamma_{1}\right|^{2}}$.

(i) WF for $\rho_{3_{a}|2}$ and $\rho_{13_{a}|2}$

$W_{\rho_{3_{a}|2}}=W_{\rho_{13_{a}|2}}=\frac{2e^{-2\left|\gamma_{2}\right|^{2}/\omega_{0}}}{\pi\omega_{0}^{3}}(4c^{2}\left|\gamma_{2}\right|^{2}-\omega_{0}),$
(B.9)

which is independent of $\phi$.

(j) WF for $\rho_{2_{a}|1}$ and $\rho_{2_{a}3|1}$

$\displaystyle W_{\rho_{2_{a}|1}}$ $\displaystyle=W_{\rho_{2_{a}3|1}}$
$\displaystyle=\frac{2e^{-2\left|\gamma_{1}\right|^{2}/\omega_{2}}}{\pi\omega_{2}^{3}}(4c^{2}\left|\gamma_{1}\right|^{2}\allowbreak\cos^{2}\phi-\omega_{2}\cos
2\phi).$ (B.10)

When $\phi=\pi/2$, Eq.(B.10) will be reduced to
$W_{\rho_{2_{a}|1}}=W_{\rho_{2_{a}3|1}}=(2/\pi)e^{-2\left|\gamma_{1}\right|^{2}}$.

(k) WF for $\rho_{1_{a}|3}$ and $\rho_{1_{a}2|3}$

$W_{\rho_{1_{a}|3}}=W_{\rho_{1_{a}2|3}}=\frac{2e^{-\allowbreak
2\left|\gamma_{3}\right|^{2}/\omega_{1}}}{\pi\omega_{1}^{3}}\allowbreak(4\left|\epsilon_{2}\gamma_{3}\right|^{2}+\omega_{1}).$
(B.11)

When $\phi=0$, Eq.(B.11) will be reduced to
$W_{\rho_{1_{a}|3}}=W_{\rho_{1_{a}2|3}}=(2/\pi)e^{-2\left|\gamma_{3}\right|^{2}}$.

(l) WF for $\rho_{2_{a}|3}$ and $\rho_{12_{a}|3}$

$\displaystyle W_{\rho_{2_{a}|3}}$ $\displaystyle=W_{\rho_{12_{a}|3}}$
$\displaystyle=\frac{2e^{-2\left|\gamma_{3}\right|^{2}/\omega_{1}}}{\pi\omega_{1}^{3}}(4c^{2}\left|\gamma_{3}\right|^{2}\allowbreak\sin^{2}\phi+\omega_{1}\cos
2\phi).$ (B.12)

When $\phi=0$, Eq.(B.12) will be reduced to
$W_{\rho_{2_{a}|3}}=W_{\rho_{12_{a}|3}}=(2/\pi)e^{-2\left|\gamma_{3}\right|^{2}}$.

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#  XPOL-III: a New-Generation VLSI CMOS ASIC for High-Throughput X-ray
Polarimetry

Minuti M  Baldini L  Bellazzini R  Brez A  Ceccanti M  Krummenacher F
Latronico L  Lucchesi L  Manfreda A  Orsini L  Pinchera M  Profeti A  Sgrò C
Spandre G  Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B.
Pontecorvo 3, I-56127 Pisa, Italy  Università di Pisa, Dipartimento di Fisica
Enrico Fermi, Largo B. Pontecorvo 3, I-56127 Pisa, Italy  Advanced Silicon SA,
Lausanne, Switzerland  Istituto Nazionale di Fisica Nucleare, Sezione di
Torino, Via P. Giuria, 1, I-10125 Torino, Italy

(Compiled on )

###### Abstract

While the successful launch and operation in space of the Gas Pixel Detectors
onboard the PolarLight cubesat and the Imaging X-ray Polarimetry Explorer
demonstrate the viability and the technical soundness of this class of
detectors for astronomical X-ray polarimetry, it is clear that the current
state of the art is not ready to meet the challenges of the next generation of
experiments, such as the enhanced X-ray Timing and Polarimetry mission,
designed to allow for a significantly larger data throughput.

In this paper we describe the design and test of a new custom, self-triggering
readout ASIC, dubbed XPOL-III, specifically conceived to address and overcome
these limitations. While building upon the overall architecture of the
previous generations, the new chip improves over its predecessors in several,
different key areas: the sensitivity of the trigger electronics, the
flexibility in the definition of the readout window, as well as the maximum
speed for the serial event readout. These design improvements, when combined,
allow for almost an order of magnitude smaller dead time per event with no
measurable degradation of the polarimetric, spectral, imaging or timing
capability of the detector, providing a good match for the next generation of
X-ray missions.

###### keywords:

X-ray polarimetry

###### PACS:

95.55.Ka , 95.55.Qf

††journal: NIM A

##  1 Introduction

The recent launches of the PolarLight cubesat [1] and the Imaging X-ray
Polarimetry Explorer (IXPE) [2] signal the re-opening of a new observational
window—that of astronomical X-ray polarimetry—after almost 40 years. At the
heart of both missions are innovative polarization-sensitive Gas Pixel
Detectors (GPD) [3], exploiting the photoelectric effect to derive the linear
polarization of the incoming radiation based on the reconstruction of the
azimuthal direction of emission of the photo-electrons. The design,
qualification and successful operation in space of the PolarLight and IXPE Gas
Pixel Detectors mark the culmination of a long R&D program [4, 5, 6],
following the first pioneering attempts at a practical implementation of
photoelectric X-ray polarimetry (see, e.g., [7]).

Future X-ray missions will pose even tighter requirements on focal-plane
detectors. With a much larger effective area (about a factor $\times~{}5$
compared to IXPE) the mirror modules of the Polarimetry Focusing Array (PFA)
onboard the enhanced X-ray Timing and Polarimetry (e-XTP) mission [8] will
produce too large of a data throughput for the current generation of Gas Pixel
Detectors, even for moderately bright sources. It is then clear that a drastic
reduction of the dead-time per event is one of the necessary preconditions for
a significant leap in sensitivity for the next generation of polarimetric
missions.

In this paper we describe XPOL-III, a new-generation custom CMOS ASIC
specifically designed for high-throughput X-ray polarimetry applications,
taking full advantage of the lessons learned through the development of the
IXPE mission. The new design is specifically aimed at a substantial reduction
of the average dead time per event, while preserving all the other relevant
high-level performance metrics of the readout chip. As detailed in section 5,
the first comprehensive test campaign that we performed on the new ASIC
confirms that all design goals were met, and XPOL-III is a prime candidate to
provide polarimetric capabilities to future X-ray missions.

##  2 XPOL-III Architecture

The main features of the previous XPOL generations, most of which are also
relevant in the context of this work, are thoroughly described in [5, 6] and
[3]. In this section we only provide a succinct summary of the internal
functioning of the chip, with emphasis on the design changes specific to XPOL-
III.

Figure 1: Picture of a XPOL-III chip (center panel), glued and wire-bonded onto its printed circuit board. The insets show micro-photographs of two of the corners at different magnification levels, with the triangular structure of the pixel array clearly visible in the bottom panel. Parameter |  Value
---|---
Number of pixels |  $107\,008~{}(304\times 352)$
Physical pitch |  50 $\mu$m
Shaping time |  1 $\mu$s
Pixel gain |  $200$ mV fC-1
Pixel Noise |  $30$ $e^{-}$ ENC
Full scale linear range (FSLR) |  $30\rm{k}~{}e^{-}$
Minimum trigger threshold |  $\sim 150$ $e^{-}$ (0.5% of FSLR)
Table 1: Summary table of the basic geometrical and electrical characteristics
of the XPOL-III readout ASIC.

Manufactured with a standard 180 nm CMOS process, the XPOL-III readout ASIC
(shown in Figure 1) is organized as a rectangular matrix of $107\,008$
hexagonal pixels arranged in a tringular pattern of $304$ columns and $352$
rows at $50$ $\mu$m pitch, for a total active area of $15.2\times 15.2$ mm2.
Each pixel is composed of a metal electrode (from the top layer of the
process), acting as a charge-collecting anode, connected to a charge-sensitive
amplifier, followed by a shaping circuit and a sample-and-hold system, as
illustrated in the simplified schematic in Figure 2.

Figure 2: Simplified pixel schematic.

Similarly to the previous ASIC generations, XPOL-III provides an advanced
self-triggering capability, with automatic localization of the _region of
interest_ (ROI) containing the photo-electron track. Upon trigger, the outputs
of the pixels within the ROI are sequentially connected to an on-chip global
differential amplifier driving the output pads, and the digitization of the
corresponding signal is performed with a dedicated, external ADC. At a very
fundamental level, this is the basic mechanism that allows to keep the overall
readout time manageable, as, for a typical event, we only read out a few
hundreds pixels out of the $>100$ k in the full matrix. Building on this
concept, in the initial stages of the XPOL-III design we identified three
distinct possible lines of actions to speed up the readout:

  * •

reduce the average size of the region of interest;

  * •

increase the maximum frequency of the serial readout clock;

  * •

streamline the readout sequence to avoid unnecessary delays.

As we shall discuss in the remaining of this section, the combination of these
three, seemingly simple updates provides a drastic dead-time reduction.

###  2.1 Event Triggering

Every 4 pixels in the ASIC are logically OR-ed together to contribute to a
local trigger with a dedicated shaping amplifier. This basic building block of
$2\times 2$ pixels is called a trigger _mini-cluster_ , and is central to the
entire trigger logic. Upon trigger, the event is automatically localized by
the ASIC core logic in the smallest rectangle containing all triggered mini-
clusters, called the _region of trigger_ (ROT). More specifically, the chip
calculates and makes available in a specific register the coordinates
$\left<X_{\text{min}},~{}Y_{\text{min}}\right>$ and
$\left<X_{\text{max}},~{}Y_{\text{max}}\right>$ of the upper-left and lower-
right corners of the ROT. On an event by event basis, this information can be
manipulated in an arbitrary fashion—typically by adding a pre-defined padding
on the four borders—to define the _region of interest_ (ROI) for the event
capture and readout.

Figure 3: Sample simulated photo-electron tracks at 8 keV (left) and 2 keV
(right) in pure DME at 800 mbar, embedded in the respective XPOL-I-like
readout windows at the nominal trigger threshold. Pixels are depicted with
black and gray borders depending on whether they lie within the ROT or the
outer padding region (8 columns on the left and the right of the ROT and 10
rows on the top and the bottom). As illustrated in the left panel, this
comparatively large padding is necessary in the XPOL-I setup in order to
guarantee that the initial part of high-energy tracks is fully contained in
the ROI, but is largely sub-optimal on the low-end of the spectrum, where
tracks are much more compact. With a significantly lower trigger threshold and
a more flexible padding strategy, XPOL-III is capable of generating much
smaller ROIs without compromising the track containment (see Figure 6).

While the top-level trigger logic in the previous generations of the ASIC was
similar to that of XPOL-III, the algorithm implemented in XPOL-I to define the
ROI was comparatively crude: a constant padding of 8 columns on the left and
right and 10 rows on the top and bottom of the ROT was automatically added by
the ASIC, as shown in Figure 3, with no room for additional tuning. This
approach served well the primary purpose of making the ROI big enough to fully
contain the tracks at all the energies of interest (and was ultimately
adequate for the typical IXPE throughput), but was clearly sub-optimal at low
energy, where tracks tend to be more compact and with a comparatively higher
ionization density.

In contrast, XPOL-III offers maximal flexibility—to the point that, once the
ROT has been defined by the ASIC, the ROI can be calculated by means of any
arbitrary external logic and loaded into the proper register on an event-
event-basis. Additionally, a _hybrid_ operational mode is available where a
pre-loaded padding, independently adjustable on the four sides of the ROT, is
automatically applied by the chip without the need of an additional data
transactions. (In practice, since the information available at the time that
the ROI decision has to be made is limited, this hybrid mode is in fact
outperforming overly sophisticated padding schemes, and will be the baseline
configuration used for all the tests described in the following of the paper.)
Even more importantly, this additional flexibility in the definition of the
ROI is accompanied by another fundamental design change: the AC nature of the
XPOL-III trigger coupling. Compared to the XPOL-I architecture, where the
dispersion of the DC offsets across different amplifiers was playing a
prominent role, this change allows to lower dramatically the minimum trigger
threshold practically achievable, increasing, in turn, the fraction of mini-
clusters in the track participating in the trigger.

It cannot be over-emphasized that these two seemingly simple changes in the
trigger circuitry constitute a change of paradigm and a radical departure from
how previous iterations of the ASIC operated. To put things in the simplest
possible way, in XPOL-I it was mostly the Bragg peak to participate into the
trigger, and we had to resort to a comparatively large padding of the ROT in
order to capture the initial part of the high-energy tracks. In XPOL-III the
entire track participates into the trigger at all energies, and a minimal
additional padding is sufficient to ensure full containment of the track. As
shown in Figure 4, this allows for ROIs that are smaller on average, and scale
more favorably with the track length, offering additional leverage for the
operation at the focus of a X-ray optics, where the vast majority of the
photons are detected at the lower end of the energy spectrum. Assuming a
constant padding of 2 (4) pixels on all four sides of the ROT, the XPOL-III
design results in an overall reduction of the ROI size by a factor of 3.5 (2),
across the band.

Figure 4: Average ROI size as a function of the X-ray energy for XPOL-I and
XPOL-III (in the latter case, with a constant padding of 2 and 4 pixels on all
four sides of the ROT). As we shall see in section 4, the shaded region
between the two solid lines is the useful phase space for determining the
optimal working point. For reference, at 2.5 keV (which is germane to the IXPE
peak sensitivity), XPOL-III yields an average ROI size comprised between 140
and 240 pixels, at the two different padding settings, which is 3.5–2 times
smaller than the XPOL-I equivalent.

###  2.2 Event Readout and Pedestal Subtraction

Upon trigger, in nominal data-taking configuration, the chip autonomously
initiates the peak-detection process of the internal sample-and-hold system,
at the end of which the analog output of each pixel within the ROI is
sequentially routed to the differential output buffer connected to the
external ADC, and the serial readout proceeds driven by an adjustable readout
clock provided by the back-end electronics. The region of interest is actually
read out twice, and the FGPA on the DAQ board then performs the pixel-by-pixel
pedestal subtraction and proceeds to the zero suppression and the event
compression and transmission [9]. As explained in [3], this online pedestal
subtraction is useful to minimize subtle systematic effects that were found to
be important for our application.

While a detailed technical description of the readout sequence is beyond the
scope of this paper, the readout time per event can be parametrized,
neglecting sub-dominant contributions, as a linear function of the number of
pixels $n_{\mathrm{p}ix}$ in the region of interest

$\displaystyle T_{\mathrm{read}}=q+mn_{\mathrm{pix}},$ (1)

where the constant term $q$ incorporates, e.g., the peak-detection interval,
the necessary data transactions to execute the readout and reset the system,
as well as all the fixed delays that are needed to synchronize the sequence,
while the slope $m$ captures all the contributions whose duration is
proportional to the size of the ROI, such as the event serial readout sequence
and the pedestal subtraction in FPGA. (We note that both $q$ and $m$ depend on
the particular settings of the readout ASIC and the back-end electronics with,
e.g., $m$ being dominated by the serial readout clock.)

Compared to XPOL-I, much of the differential output has been redesigned in
XPOL-III in order to streamline the readout. The design of the output buffer
has been tuned to minimize the settling time of the analog signal and increase
the maximum serial readout clock practically usable, which for XPOL-I was
limited to $\sim 5$ MHz, and the analog reset circuitry has been reworked to
avoid the need for extra delays, necessary in XPOL-I to avoid self-induced
false triggers. One additional notable change is that the ROI coordinates,
once determined, remain stored in their register until the latter is
explicitly reset, which avoids another class of fixed delays necessary in
XPOL-I to perform the online pedestal subtraction. The net result of all these
optimization is a net (and significant) reduction of both terms in Equation
(1), as we shall see in section 4.

##  3 Measurement setup

The GPD assembly is the natural _packaging_ for the readout ASIC, and an
essential ingredient for testing its functionality beyond the basic electrical
aliveness. We have assembled, baked-out, filled and sealed entirely in house,
a few detectors equipped with a XPOL-III readout ASIC. Figure 5 shows an
exploded view of the detector, whose design implements some important
simplification with respect to the focal plane detectors onboard IXPE [3].

Figure 5: Exploded view of a GPD equipped with a XPOL-III readout ASIC. The
main differences with respect to the models used in IXPE are the chip-on-board
configuration and the absence of the internal alumination of the Be foil.

We kept the top-level geometry and gas mixture, but we opted for a chip-on-
board scheme (i.e. with the ASIC directly glued and wire-bonded over its
printed circuit board, avoiding the use of a ceramic package) to simplify the
assembly. We used $50~{}\mu$m pitch, $50~{}\mu$m thick GEM foils manufactured
with a conventional chemical etching process. Finally, the entrance window is
a simple $50~{}\mu$m Be foil with no alumination. The main characteristics of
the detectors are summarized in table 2.

Parameter |  Value
---|---
Entrance window |  Pure Be, 50 $\mu$m
Drift gap thickness |  10 mm
Transfer gap thickness |  0.8 mm
Readout configuration |  Chip on board
GEM pitch |  50 $\mu$m
GEM hole diameter |  30 $\mu$m
Gas filling |  Pure DME @ 730 mbar
Table 2: Summary table of the basic properties of the GPD under test.

Each one of these modifications is a step toward our ultimate goal of a new-
generation GPD, featuring a simpler and more robust mechanical assembly, and a
higher polarization sensitivity—both in terms of modulation factor and
uniformity of response to unpolarized radiation. A detailed description of the
GPD development, including the description of a dedicated filling facility and
the long-term stability of the performance, is beyond the scope of this work
and will be presented in a forthcoming paper.

##  4 Electrical tests and working point definition

Figure 6: Display of a real XPOL-III track from a 5.9 keV radioactive 55Fe
source with a padding of three pixels on all four sides of the ROT. The
indices on the top and left size of the ROI show the location on the readout
matrix, while the text overlaid on the pixels represent the corresponding PHA
value. Even a superficial comparison with Figure 3 reveals how, given the much
lower trigger threshold, the vast majority of the track is now contained in
the ROT, and the padding of three pixels used here is in fact very generous
even for a typical high energy track.

This section describes the initial electrical and functional tests that we
performed to verify the basic functionality of the ASIC and gauge the optimal
working point, in terms of event padding and serial readout clock, to be used
for measuring the relevant high-level performance metrics that we shall
present in section 5. For completeness, Figure 6 shows a single event display
of a real track from a 5.9 keV radioactive 55Fe source, illustrating much of
the improvements from the new trigger circuitry discussed in section 2.

###  4.1 Electronics Noise

As already explained, and unlike the previous XPOL generations, XPOL-III
allows to set the coordinates of the ROI for the readout externally, and we
extensively used this novel feature to measure the system noise in a number of
readout chips. More specifically, we use a series of partially overlapping,
$21\times 21$ regions of interest covering the entire active area of the chip
and, for each ROI, we perform a number of readouts, with the same basic
readout sequence used in nominal data-taking. For each pixel, the root mean
square of the pedestal subtracted PHA values is a measure of the pre-amplifier
noise111To be more precise, since we are operating with an online, event-by-
event pedestal subtraction strategy as described in section 2, the raw rms
values are a factor of $\sqrt{2}$ larger than the intrinsic noise of the
amplifier. This scale factor is correctly accounted for in Figure 7 as well as
in the numbers quoted in the text.

Figure 7: Noise distribution measured across the channels of a typical XPOL-
III chip. The average noise is $\sim 30~{}e^{-}$ ENC.

Figure 7 shows the noise distribution measured across the channels of a
typical XPOL-III chip. The average noise is about $30~{}e^{-}$ ENC. It should
be emphasized that this level of noise, translating into a track S/N ratio of
$\sim 200$ for a typical event, is largely irrelevant for our applications,
and not a limiting factor in any practical sense.

We note that the initial chips manufactured at the foundry and tested for this
work have a small fraction of a ‰ dead channels; while this does not represent
an issue for operating a GPD, a fine tuning of the manufacturing will be
investigated with the foundry.

###  4.2 ROT Padding

We systematically investigated the effect of the ROT padding settings on the
basic track properties by exploiting the hybrid XPOL-III operational mode
described in section 2 and acquiring X-ray data with different, pre-loaded
padding values—from 2 to 5 pixels on all four sides of the ROT. Since the main
purpose of the padding is to guarantee the full containment of the electron
track, we resorted to using _the fraction of events for which at least one of
the pixels in the track lies on a border row or column_ , which might indicate
a possible track leakage, as the relevant metric for our study.

Figure 8: Fraction of events for which at least one of the pixels in the main
track lies on the borders of the ROI, as s function of the ROT padding
setting. The data points have been measured with a full irradiation of the
detector with 5.9 keV photons from a 55Fe radioactive source, with the zero
suppression threshold set to $\sim 2\sigma$ of the noise.

As shown in Figure 8 the fraction of tracks with border pixels decreases
monotonically as the padding increases, going from several % at 2 pixels to a
few $10^{-5}$ at 5 pixels. We emphasize that a single pixel above threshold on
one of the borders of the ROI does not necessarily imply that the track has
been actually truncated, nor that the polarimetric information encoded by the
reconstructed track direction has been compromised appreciably. We therefore
conservatively take a padding value of 2 pixels as the smallest practically
usable and we deem padding values larger than 4 pixels as overly large, as the
plot in Figure 8 clearly indicates that the curve starts flattening (cf.
Figure 4).

We note that more elaborated padding strategies (e.g., use different values on
different sides of the ROT) might conceivably provide an additional measurable
performance gain. The magnitude of the potential improvement can be gauged by
noting that reducing the padding by one unit on one side provides a relative
reduction of the ROI size by

$\displaystyle\frac{\delta n}{n}\approx\frac{1}{\sqrt{n}},$ (2)

which is $\sim 7\%$ at a typical average ROI size of 200 pixels. Since
investigating the effects of such fine tuning is beyond the scope of this
paper, we shall take a constant, comfortable padding of $3$ pixels as our
nominal working point from now on, unless stated otherwise.

###  4.3 Readout Time

As explained in section 2.2, the average readout time per event can be
parametrized as a linear function of the number of pixels in the ROI. Since,
for any given X-ray energy, events come in a large variety of different
topologies, depending on the depth of the absorption point and the initial
emission direction, as well as the stochastic nature of the interaction
processes-even photons from a monochromatic source will generate in the
detector ROIs of many different sizes and shapes. As a consequence,
illuminating the detector with a X-ray source provides a simple mean to study
the dead time per event as a function of the ROI size.

Figure 9: Readout time as a function of the ROI size, for two different values
of the serial readout clock, obtained by illuminating the detector with 5.9
keV photons from a 55Fe radioactive source, with a padding of 3 pixels on all
four sides of the ROT. The solid lines represent the best linear fits to the
data points for two different values (6 and 10 MHz) of the serial readout
clock. The vertical gray band indicates the average ROI size that we expect at
the reference energy of 2.5 keV for a padding of 2 and 4 pixels. The hatched
region between the two dashed lines indicate the intrinsic XPOL-III readout
limit (at 6 and 10 MHz readout clock), assuming that the pedestal subtraction
is fully parallelized and in the ideal case where the back-end electronics did
not introduce any additional delay to the readout sequence.

Figure 9 shows the measured readout time as a function of the ROI size for two
different values of the serial readout clock (6 and 10 MHz), and a constant
padding of 3 pixels on all four sides of the ROT, with the detector uniformly
illuminated with 5.9 keV X-rays from a 55Fe radioactive source. The data are
fitted with the straight line in Equation (1). At our reference energy of
$\sim 2.5$ keV, the center of our fiducial region in the padding vs readout
clock phase-space yields a tentative working point of $\sim 150~{}\mu$s, at an
average ROI size of $\sim 200$ pixels. This is to be compared with $\sim 500$
pixels and $\sim 1$ ms for the XPOL-I chip currently operating on PolarLight
and IXPE.

We note that, due to the current design of our back-end electronics, the
measured readout time includes a contribution of $\sim 35~{}\mu$s for the
pedestal subtraction in the FPGA, for a ROI of 200 pixels. For a high-
throughput application, this component could be conceivably parallelized at
the cost of a slight increase in the complexity of the system, yielding an
additional $>20$% increase in readout speed without side effects on any of the
high-level performance metrics.

##  5 High-Level Performance Figures

In this section we briefly discuss the basic spectral and polarimetric
response of the XPOL-III chip, embedded in the new GPD design. The reader
should keep in mind that the readout ASIC is only a component of the mix, and
the results are dependent on a number of other factors, including the detector
assembly, the characteristic of the filling gas, the amplification stage, as
well as the track-reconstruction software. For the sake of internal
consistency, all the tests described in the following are done at the working
point identified in section 4, that is, with a constant padding of 3 pixels on
the four sides of the ROT, and a serial readout clock of 7.5 MHz.

###  5.1 Energy Resolution

Figure 10 shows the pulse-height distribution in pure DME at the nominal data
taking settings for a flat field with 5.9 keV photons from a 55Fe radioactive
source. After correcting for spatial non-uniformities of the GEM, we achieve a
FWHM around 17%, in line with the corresponding figure for the IXPE flight
detectors [3].

Figure 10: Spectrum of 55Fe line with a gaussian fit in nominal working
conditions. The detector surface is uniformly illuminated and the large-scale
gain dis-uniformity of the Gas Electron Multiplier corrected.

We emphasize how even a superficial comparison with Figure 16 in [3] shows
that the low-energy tail in the spectrum due to photon conversions in the
passive materials is reduced by a factor $\sim 2$, due to the elimination of
the aluminum deposition on the inner face of the entrance window. While the
suitability of a pure Be window for a long-term use in space will be
investigated and presented in a separate paper, we note that this factor is
beneficial for both the spectral deconvolution and the polarization
measurement.

###  5.2 Azimuthal Response

We tested the azimuthal response of the GPD assembly with both unpolarized
photons from a 5.9 keV 55Fe radiactive source and polarized X-rays generated
via Bragg diffraction at 45∘ on a graphite crystal. Our polarized setup
produces three different monochromatic lines at 2.6, 5.2 and 7.8 keV, but the
low- and high-energy ones are strongly suppressed due to the absorption in air
and the energy dependence of the GPD quantum efficiency, respectively.

Figure 11: Measured modulation curve at 5.2 keV for a 100% polarized X-ray
source. The modulation factor is 46.7% without any selection on the events, in
line with previous measurements and with the prediction of our Monte Carlo
simulation.

Figure 11 shows two modulation curves, measured in our unpolarized and
polarized setups. For the latter, we selected events in the central 5.2 keV
over a beam spot with a $\sim 1$ mm radius close to the center of the
detector, and performing two separate acquisitions rotated by 90∘ in order to
compensate for any possible residual systematic effect. (Although, as shown in
the bottom panel, such effects are largely subdominant.) The measured
modulation factor at 5.2 keV is $46.7\pm 0.5$ %, without any additional
selection on the events other than that of the central line in the source
spectrum. The measured spurious modulation at 5.9 keV is smaller than 1%, and,
as explained in [3], is to be ascribed to the multiplication stage. This is in
good agreement with both the figures for the IXPE focal-plane detectors (given
the differences in the GPD assembly and the X-ray source) and with the
prediction from our Monte Carlo simulations, indicating that the new readout
chip is fully preserving the intrinsic polarimetric capabilities of the GPD.

##  6 Conclusions

We have developed and tested a new custom readout chip for high-throughput
X-ray astronomical polarimetry. Compared to the ASIC currently operating on
PolarLight and IXPE, XPOL-III is able to generate significantly smaller, and
yet fully contained track images and to operate at a faster readout clock,
with a streamlined readout sequence.

When combined, these factors reduce the average deadtime per event to $\sim
150~{}\mu$s, or a factor of 7 smaller than the current state of the art.
Considering that at least $\sim 35~{}\mu$s of the measured deadtime are an
overhead introduced by the existing back-end electronics, a speed-up of a full
order of magnitude is clearly within reach, making XPOL-III an ideal match for
the upcoming generation of X-ray observatories.

This leap forward in readout speed is achieved with no measurable degradation
in the polarimetric, spectral, imaging or timing capability of the detector,
as demonstrated by the tests presented in this work.

In addition, the reduction of at least a factor 2.5 of the ROI size is clearly
very helpful in reducing the required bandwidth for both on board satellite
operation as well for data transmission to the ground.

Finally, the possibility of operating at a trigger threshold as low as $\sim
150~{}e^{-}$ paves the way to new types of applications that were previously
impossible, e.g., low-pressure configurations tailored to very low energies
or, on the other side of the energy band, operating in pure ionization mode
with suitable gas mixtures.

We argue that this work constitutes a crucial step toward the deployment of a
new generation of GPD matching the needs of future X-ray observatories. The
development of suitable multiplications stages, free of systematic effects in
the azimuthal response, is the other fundamental ingredient of the mix, and we
shall report our progress in a separate paper.

##  7 Acknowledgements

This work was supported by the Italian Space Agency (ASI) through the
agreements 2018-11-HH.O, “ADAM - Advanced Detectors for X-ray Astronomy
Mission” and 2017.13-H0, ”Italian participation to the NASA IXPE mission”

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