11institutetext: Instituto de Astronomía Teórica y Experimental, CONICET-UNC,
Córdoba, Argentina. 22institutetext: Facultad de Matemática, Astronomía,
Física y Computación, Universidad Nacional de Córdoba (UNC), Córdoba,
Argentina. 33institutetext: Observatorio Astronómico de Córdoba, UNC, Córdoba,
Argentina. 44institutetext: Centre for mathematical Plasma Astrophysics,
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven,
Belgium 55institutetext: Instituto de Estudios Avanzados en Ingeniería y
Tecnología, CONICET, Córdoba, Argentina. 66institutetext: Facultad de Ciencias
Exactas, Físicas y Naturales, UNC, Córdoba, Argentina. 77institutetext: Grupo
de Estudios en Heliofísica de Mendoza, CONICET, Universidad de Mendoza,
Mendoza, Argentina.

# Pseudostreamer influence on flux rope evolution

A. Sahade 112233 M. Cécere 1133 M.V. Sieyra 44 G. Krause 5566 H. Cremades 77
A. Costa 11

(Received ; accepted )

###### Abstract

Context. A highly important aspect of solar activity is the coupling between
eruptions and the surrounding coronal magnetic field topology, which
determines the trajectory and morphology of the event. Pseudostreamers (PSs)
are coronal magnetic structures formed by arcs of twin loops capped by
magnetic field lines from coronal holes of the same polarity that meet at a
central spine. They contain a single magnetic null point in the spine, just
above the closed field lines, which potentially influences the evolution of
nearby flux ropes (FRs).

Aims. Because of the impact of magnetic FR eruptions on space weather, we aim
to improve current understanding on the deflection of CMEs. To understand the
net effect of the PS on FR eruptions is first necessary to study diverse and
isolated FR-PS scenarios, which are not influenced by other magnetic
structures.

Methods. We performed numerical simulations in which a FR structure is in the
vicinity of a PS magnetic configuration. The combined magnetic field of the PS
and the FR results in the formation of two magnetic null points. We evolve
this scenario by numerically solving the magnetohydrodynamic equations in
2.5D. The simulations consider a fully ionised compressible ideal plasma in
the presence of a gravitational field and a stratified atmosphere.

Results. We find that the dynamic behaviour of the FR can be categorised into
three different classes based on the FR trajectories and whether it is
eruptive or confined. Our analysis indicates that the magnetic null points are
decisive in the direction and intensity of the FR deflection and their
hierarchy depends on the topological arrangement of the scenario. Moreover,
the PS lobe acts as a magnetic cage enclosing the FR. We report that the total
unsigned magnetic flux of the cage is a key parameter defining whether the FR
is ejected or not.

###### Key Words.:

Sun: coronal mass ejections (CMEs) – Magnetohydrodynamics (MHD) – Methods:
numerical

##  1 Introduction

Magnetic flux ropes (FRs) are thought to be the central structures of solar
eruptions, including prominence/filament eruptions, flares, and coronal mass
ejections (CMEs). The physical process connecting these phenomena is the
eruption of the magnetic flux rope system (Zhang et al.,, 2001; van Driel-
Gesztelyi and Green,, 2015; Green et al.,, 2018; Jiang et al.,, 2018; Yang et
al.,, 2018; Filippov,, 2019). Knowing whether the FR will erupt or not is,
naturally, fundamental to predicting a CME event. Gronkiewicz et al., (2016)
pointed out several mechanisms that can decelerate and confine eruptions in
the corona. The first one is the action of gravity, which prevents the
eruption when the energy of the FR is not enough to escape the gravitational
potential of the Sun (Filippov,, 2021). Even if the FR has the energy to
escape gravity, the eruption would be confined if the overlying arcade field,
whose lines form a magnetic cage, is too strong or do not quickly decay with
height (Török and Kliem,, 2005; Wang and Zhang,, 2007; Chen et al.,, 2013;
Amari et al.,, 2018; Baumgartner et al.,, 2018; Jing et al.,, 2018). Recently,
Li et al., (2020, 2021, 2022) proved that there exists a negative correlation
between the flare eruptivity (i.e. if it has an associated CME) and the total
unsigned magnetic flux of the active region producing the flare, which
describes the strength of the background field confinement.

Once the eruption occurs, it is essential to understand the path that a CME
will follow in order to predict its geoeffectiveness. This requires knowledge
of any non-radial propagation of the CME, for which deflections in the
trajectory must be studied. It is widely known that the magnetic structures in
the vicinity of FRs are capable of deflecting them both in latitude and
longitude. While coronal holes (e.g., Cremades et al.,, 2006; Gopalswamy et
al.,, 2009; Sahade et al.,, 2020, 2021) and active regions (e.g., Kay et al.,,
2015; Möstl et al.,, 2015; Wang et al.,, 2015) deflect FRs against their
location, heliospheric current sheets (e.g., Liewer et al.,, 2015; Wang et
al.,, 2020), helmet-streamers (e.g., Zuccarello et al.,, 2012; Yang et al.,,
2018) and pseudostreamers (PSs) (e.g., Bi et al.,, 2013; Wang,, 2015; Cécere
et al.,, 2020; Wang et al.,, 2020) attract FRs to their low magnetic energy
regions. Combined effects of the several structures at different heights can
be seen in, for example, Sieyra et al., (2020). In previous studies (Sahade et
al.,, 2020, 2021), we found that the presence of a coronal hole nearby the
eruptive region forms a magnetic null point that attracts the FR. The null
point can be located between the FR and CH (case of anti-aligned polarities)
or at the other side of the FR (case of aligned polarities). The first
scenario produces an initial deflection towards the CH and a second deflection
against its position. On the other hand, the aligned polarities cases lead to
a single deflection in which the FR moves away from the CH. All the final
paths are opposite to the location of the coronal hole by the “channelling” of
the magnetic field lines, i.e., the FR is guided to follow the least
resistance path. Möstl et al., (2015) and Wang et al., (2015) studied an event
on 2014 January 7 whose deflection seems to be caused by the magnetic pressure
gradient from a nearby active region and whose final path is also channelled
by the configuration of the magnetic field lines to the least resistance
direction. Shen et al., (2011) concluded that the trajectory in the early
stages is influenced by the background magnetic energy gradients, inducing the
CME to propagate towards the region with the lowest magnetic energy density.
Similar results were found by Sieyra et al., (2020) where most of the analysed
CMEs eruptions were aligned with the direction of the magnetic energy
decrease. It also showed that most of the deflection occurs at heights lower
than 2.4 $R_{\odot}$, suggesting that it is of utmost importance to study the
trajectory in the early stage.

Streamers are characterised by containing a region of null magnetic energy,
therefore they can act as a potential well attracting CMEs towards them (Kay
et al.,, 2013). In particular, the PS contains a single magnetic null point
above the closed field lines. These closed field lines that overlie two (or an
even number of) polarity inversion lines, are covered by open field lines of
the same polarity, without a current sheet, forming the spine of the
pseudostreamer (Rachmeler et al.,, 2014). Observational studies have suggested
that there is a null point hierarchy: the rolling motions and deflections of
prominences are caused by the nearest local null point and the CMEs move in a
non-radial direction towards the global null point located at higher altitudes
associated to helmet streamers or pseudostreamers (Panasenco et al.,, 2013).
In order to explain the physical processes involved in the deflection of the
eruptive phenomenon there are some numerical studies that analyse the CME
deflections in presence of PS structures. For example, Zuccarello et al.,
(2012) found, in a scenario where the heliospheric current sheet and PS are
both present, a CME that erupts from one of the PS lobes and is firstly
deflected towards the null point of the PS and then continues moving towards
the heliospheric current sheet. A similar behaviour is found in the simulation
performed by Wyper et al., (2021), in which the PS is embedded in a helmet
streamer. Recently, Karna et al., (2021) modelled the eruptive filament
observed on 2015 April 19, which was embedded in a lobe of the pseudostreamer
and was directed towards the PS null point. Although it is well established,
both numerically and observationally, that FRs inside PS deflect towards the
PS spine (e.g., Török et al.,, 2011; Zuccarello et al.,, 2012; Yang et al.,,
2015; Karna et al.,, 2021), to date there are no studies that analyse how the
trajectory of a FR is affected by variations in the FR-PS configuration.

In this paper, we model cases where only one FR interacts with the PS
structure, and we analyse its influence on the FR trajectory at low coronal
heights by 2.5 MHD numerical simulations. In Section 2, we describe the
numerical model details and parameters for the presented cases. In Section 3,
we present results arising from the several simulations performed. We observed
that the FR-PS interactions can be distinguished into three separate classes
that exhibit differences in the dynamic behaviour of the FR. On one hand,
differences in the magnetic field topology lead to different hierarchies of
the null points and consequently changes in the dynamical behaviour. On the
other hand, we found that the eruption or confinement of the FR strongly
depends on the unsigned magnetic flux of the magnetic cage. Discussion and
final remarks are presented in Section 4.

##  2 Numerical simulations

To study the interaction between a FR and a PS, we present a scenario where
both structures interact in isolation. In this way, we avoid the possible
effects of other magnetic structures that could affect the FR behaviour,
allowing a comprehensive analysis of the PS influence in the FR evolution. We
consider the ideal MHD equations in presence of a gravitational field to solve
the 2.5 dimensional model. In CGS units in the Cartesian conservative form we
have:

$\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\@vec{v})=0$
$\displaystyle(continuity)$ (1)
$\displaystyle\frac{\partial(\rho\@vec{v})}{\partial
t}+\nabla\cdot\left(\rho\@vec{v}\@vec{v}-\frac{1}{4\pi}\@vec{B}\@vec{B}\right)+\nabla
p+\nabla\left(\frac{B^{2}}{8\pi}\right)=\rho\@vec{g}$
$\displaystyle(momentum)$ (2) $\displaystyle\frac{\partial E}{\partial
t}+\nabla\cdot\left[\left(E+p+\frac{B^{2}}{8\pi}\right)\@vec{v}-\frac{1}{4\pi}\left(\@vec{v\cdot
B}\right)\@vec{B}\right]=\rho\@vec{gv}$ $\displaystyle(energy)$ (3)
$\displaystyle\frac{\partial\@vec{B}}{\partial
t}+\@vec{\nabla\cdot}\left(\@vec{v}\@vec{B}-\@vec{B}\@vec{v}\right)=\@vec{0}$
$\displaystyle(induction)$ (4)

where $\rho$ represents the plasma density, $p$ the thermal pressure,
$\@vec{v}$ the velocity, $\@vec{B}$ the magnetic field, and $\@vec{g}$ the
gravity acceleration. $E$ is the total energy (per unit volume), given by

$E=\rho\epsilon+\frac{1}{2}\rho v^{2}+\frac{B^{2}}{8\pi},$

where $\epsilon$ is the internal energy and

${\@vec{j}=\frac{c}{4\pi}{\nabla\times}\@vec{B},}\,$

is the current density, with $c$ being the speed of light.

In addition to the MHD equations, the divergence-free condition of the
magnetic field must be fulfilled, i.e.

$\@vec{\nabla\cdot}\@vec{B}=0\,.$ (5)

We assume that the medium is a fully ionised hydrogen plasma, for which is
valid the perfect gas law $p=2\rho k_{B}T/m_{i}=(\gamma-1)\rho\epsilon$.
$k_{B}$ is the Boltzmann constant, $T$ the plasma temperature, $m_{i}$ the
proton mass, and $\gamma=5/3$ the specific heat relation.

Simulations were performed using the FLASH Code (Fryxell et al.,, 2000) in its
fourth version, operated under an adaptive refinement mesh with the USM
(Unsplit Staggered Mesh) solver, which uses a second-order directionally
unsplit scheme with a MUSCL-type (Monotonic Upstream-centered Scheme for
Conservation Laws) reconstruction. We use the local Lax-Friedrichs Riemann
solver, which is a diffusive solver providing the necessary dissipation to
emulate the magnetic resistivity and use the ideal MHD equations (Sahade et
al.,, 2020). Outflow conditions (zero-gradient) are used at lateral and upper
boundaries, while the line-tied condition is used at the lower boundary, which
imposes the condition of null velocity and constant magnetic field for the
ghost cells (Robertson and Priest,, 1987). In the guard-cells of the boundary
the magnetic field is linearly extrapolated to preserve the divergence-free
configuration. The highest resolution corresponds to $\sim[0.1\times
0.1]\leavevmode\nobreak\ \mathrm{Mm}^{2}$ cells, in a
$[-700,700]\leavevmode\nobreak\ \mathrm{Mm}\times[0,700]\leavevmode\nobreak\
\mathrm{Mm}$ physical domain, where pressure and temperature gradients satisfy
the refinement criterion.

###  2.1 FR and PS magnetic model

The modelling of the FR magnetic structure is based on the catastrophe model
by Forbes, (1990) consisting of an out-of equilibrium magnetic configuration
that triggers the FR ejection. The model of the PS is based on the magnetic
configuration proposed by Edmondson et al., (2010). However, to better
reproduce the decay of the magnetic field with altitude in the solar corona,
we replace the constant background magnetic field in the $y$-direction by an
exponentially decaying field. The $x$-direction is oriented along the
horizontal coordinate, the $y$-direction corresponds to the vertical
coordinate and the $z$-direction is the direction of symmetry. Combining both
models, the total magnetic field is given by:

$\displaystyle B_{x}$ $\displaystyle=B_{x,\mathrm{FR}}+B_{x,\mathrm{PS}}\,,$
$\displaystyle B_{y}$ $\displaystyle=B_{y,\mathrm{FR}}+B_{y,\mathrm{PS}}\,,$
$\displaystyle B_{z}$ $\displaystyle=B_{z,\mathrm{FR}}\,.$

The magnetic field components of the FR are given by the sum of a current
wire, an image current wire and a line dipole:

$\displaystyle B_{x,\mathrm{FR}}=-$ $\displaystyle
B_{\phi}(R_{-})\tfrac{(y-h_{0})}{R_{-}}+B_{\phi}(R_{+})\tfrac{(y+h_{0})}{R_{+}}-$
$\displaystyle
MdB_{\phi}{\scriptstyle\left(r+\tfrac{\Delta}{2}\right)}\left(r+\tfrac{\Delta}{2}\right)\tfrac{x^{2}-(y+d)^{2}}{R_{d}^{4}}\,,$
$\displaystyle B_{y,\mathrm{FR}}=\quad$ $\displaystyle
B_{\phi}(R_{-})\tfrac{x}{R_{-}}-B_{\phi}(R_{+})\tfrac{x}{R_{+}}-$
$\displaystyle
MdB_{\phi}{\scriptstyle\left(r+\tfrac{\Delta}{2}\right)}\left(r+\tfrac{\Delta}{2}\right)\tfrac{2x(y+d)}{R_{d}^{4}}\,,$
$\displaystyle B_{z,\mathrm{FR}}=\quad$ $\displaystyle B_{\text{z}}(R_{-})\,.$
(6)

In these expressions, $h_{0}$ is the initial height of the FR, $M$ is the
intensity of the line dipole at depth $d$, $r$ is the current wire radius,
$\Delta$ is the thickness of the transition layer between the current wire and
the exterior, and $R_{\pm}=\sqrt{x^{2}+(y\pm h_{0})^{2}}$ and
$R_{d}=\sqrt{x^{2}+(y+d)^{2}}$ are the distances taken from different origins
(image and current wire, and dipole, respectively). Also,

$B_{\phi}{(R)}\\!=\\!\left\\{\begin{array}[]{rl}\begin{aligned}
&\tfrac{2\pi}{c}j_{0}R&&0\leq R<r-\frac{\Delta}{2}\\\ &\tfrac{2\pi
j_{0}}{cR}\left\\{\tfrac{1}{2}\left(r-\tfrac{\Delta}{2}\right)^{2}-\left(\tfrac{\Delta}{2}\right)^{2}+\right.\\\
&\tfrac{R^{2}}{2}+\tfrac{\Delta
R}{\pi}\text{sin}\left[\tfrac{\pi}{\Delta}\left(R-r+\tfrac{\Delta}{2}\right)\right]+\qquad&&r-\frac{\Delta}{2}\\!\leq\\!R\\!<r+\frac{\Delta}{2}\\\
&\left.\\!\\!\\!\left(\tfrac{\Delta}{\pi}\right)^{2}\cos\left[\tfrac{\pi}{\Delta}\left(R-r+\tfrac{\Delta}{2}\right)\right]\right\\}\\\
&\tfrac{2\pi
j_{0}}{cR}\left[r^{2}+\left(\tfrac{\Delta}{2}\right)^{2}-2\left(\tfrac{\Delta}{\pi}\right)^{2}\right]&&r+\frac{\Delta}{2}\leq
R\end{aligned}\end{array}\right.$ (7)
$j_{z}{(R)}\\!=\\!\left\\{\begin{array}[]{rl}\begin{aligned} &\\!j_{0}&&0\leq
R<r-\frac{\Delta}{2}\\\
&\\!\tfrac{j_{0}}{2}\left\\{\cos\left[\tfrac{\pi}{\Delta}\left(R-r+\tfrac{\Delta}{2}\right)\right]+1\right\\}\quad&&r-\frac{\Delta}{2}\\!\leq\\!R\\!<r+\frac{\Delta}{2}\\\
&\\!0&&r+\frac{\Delta}{2}\leq R\end{aligned}\end{array}\right.$ (8)

where $j_{0}$ is a current density. The component $B_{\mathrm{z}}$ of the
magnetic field and the current distribution $j_{\phi}$, are described by:

$B_{\mathrm{z}}(R)=\tfrac{4\pi
j_{1}}{c}\sqrt{\left(r-\tfrac{\Delta}{2}\right)^{2}-R^{2}}\,,$ (9)
$j_{\phi}(R)=j_{1}R\left[\sqrt{\left(r-\tfrac{\Delta}{2}\right)^{2}-R^{2}}\right]^{-1}\,,$
(10)

where $j_{1}$ is a current density. These expressions are valid in $0\leq
R<r-\frac{\Delta}{2}$ and are null in the rest of the domain.

The magnetic field components of the PS are composed by a line dipole and a
potential field:

$\displaystyle B_{x,\mathrm{PS}}(x,y)=$ $\displaystyle\frac{2\sigma
B_{\text{PS}}(x-x_{\text{PS}})(y-y_{\text{PS}})}{((x-x_{\text{PS}})^{2}+(y-y_{\text{PS}})^{2})^{2}}+$
(11a) $\displaystyle B_{0}\
\sin\left(\frac{x-x_{\text{PS}}}{H}\right)\,\exp[-y/H]\,,$ $\displaystyle
B_{y,\mathrm{PS}}(x,y)=$ $\displaystyle-\frac{2\sigma
B_{\text{PS}}(x-x_{\text{PS}})^{2}}{((x-x_{\text{PS}})^{2}+(y-y_{\text{PS}})^{2})^{2}}+$
(11b) $\displaystyle\frac{\sigma
B_{\text{PS}}}{(x-x_{\text{PS}})^{2}+(y-y_{\text{PS}})^{2}}+$ $\displaystyle
B_{0}\ \cos\left(\frac{x-x_{\text{PS}}}{H}\right)\,\exp[-y/H]\,,$

where $\sigma B_{\mathrm{PS}}$ is the strength of the magnetic field due to a
single line dipole ($\sigma=2\times 10^{21}$ is a dimensionless scale factor)
positioned at $(x,y)=(x_{\mathrm{PS}},y_{\mathrm{PS}})$, $B_{0}$ is the
strength of the background field at $(x,y)=(x_{\mathrm{PS}},0)$, and $H=600$
Mm is the height decaying factor.

Figure 1(a) shows a scheme with the distribution of the magnetic structures
and 1(b) the internal structure of the FR. In the left panel, the green frame
(top) represents the simulated domain and the grey shaded area (bottom)
contains the magnetic components that are out of the simulation box. The PS
model produces a four-flux system, the separation between these regions (red
lines) is characterised by a magnetic null point (red circle). Two of the
fluxes have a closed topology (red shaded area) and form the PS lobes that are
divided by the spine (vertical red line). Outside this region , delimited by
the semicircular red line, two open fluxes of equal polarity converge towards
the spine and surround the PS structure.

Figure 1: (a) Flux rope (orange circle) and pseudostreamer (red structure)
scheme, not to scale. The green frame indicates the simulated region. $h_{0}$
is the FR height and $x_{\mathrm{PS}}$ is the distance from the FR to the PS
spine (red vertical line), whose height is parametrised by $y_{\mathrm{PS}}$.
The line dipole and the image current are located at depth $d$ and $h_{0}$,
respectively. (b) Internal structure of the FR: $r$ is the radius, $\Delta$ is
the thickness of the transition layer, $R_{-}$ is the radial coordinate from
the FR centre, and $\phi$ the poloidal coordinate.

###  2.2 Thermodynamic variables

We simulate the solar atmosphere by adopting a multi-layer structure (Mei et
al.,, 2012). The chromosphere lies between the heights $0\leq y\leq
h_{\mathrm{ch}}=10\,$Mm with constant temperature
$T_{\mathrm{ch}}=10\,000\,\text{K}$. The base of the corona is at a height
$h_{\mathrm{c}}=15\,\text{Mm}$ and has constant temperature
$T_{\mathrm{c}}=10^{6}\,\text{K}$. These layers are connected by the
transition region, which has a linearly increasing temperature. Thereby, the
initial temperature distribution is given by

${\textstyle T(y)=}\left\\{\begin{array}[]{rl}\begin{aligned} &{\textstyle
T_{\mathrm{ch}}}&&0\leq y<h_{\mathrm{ch}}\\\
&{\textstyle(T_{\mathrm{c}}-T_{\mathrm{ch}})\left[\frac{y-h_{\mathrm{ch}}}{h_{\mathrm{c}}-h_{\mathrm{ch}}}\right]+T_{\mathrm{ch}}}\quad&&h_{\mathrm{ch}}\leq
y<h_{\mathrm{c}}\\\ &{\textstyle T_{\mathrm{c}}}&&h_{\mathrm{c}}\leq
y.\end{aligned}\end{array}\right.$ (12)

The temperature inside the FR ($T_{\text{\tiny{FR}}}$) varies according to the
following temperature distribution:

$\\!T(R_{-})\\!=\\!\left\\{\begin{array}[]{rl}\begin{aligned}
&\\!T_{\text{\tiny{FR}}}&&0\leq R<r-\frac{\Delta}{2}\\\
&\\!\\!(T_{\mathrm{c}}\\!-\\!T_{\text{\tiny{FR}}})\\!\left[\tfrac{R_{-}-(r+\Delta/2)}{\Delta}\right]\\!+\\!T_{\text{\tiny{FR}}}\quad&&r-\frac{\Delta}{2}\\!\leq\\!R\\!<r+\frac{\Delta}{2}\\\
&\\!T_{\mathrm{c}}&&r+\frac{\Delta}{2}\leq R.\end{aligned}\end{array}\right.$
(13)

We consider a current-free atmosphere in hydrostatic equilibrium. Hence, the
background pressure $p(y)$ is only a function of height considering a system
having the $y$-axis aligned to the gravity acceleration (i.e.,
$\@vec{g}=\frac{-GM_{\sun}}{(y+R_{\sun})^{2}}\@vec{e}_{y}$, where $G$ is the
gravitational constant, $M_{\sun}$ is the solar mass, $R_{\sun}$ is the solar
radius, and $y=0$ corresponds to the solar surface). Therefore, the
atmospheric pressure is:

${\textstyle p(y)=\\!}\left\\{\begin{array}[]{rl}\begin{aligned}
&{\textstyle\\!\\!p_{\mathrm{ch}}\exp{\\!\left[\frac{\alpha}{T_{\mathrm{ch}}}\left(\frac{1}{h_{\mathrm{ch}}+R_{\sun}}-\frac{1}{y+R_{\sun}}\right)\right]}}&&0\leq
y<h_{\mathrm{ch}}\\\
&{\textstyle\\!\\!p_{\mathrm{ch}}\exp{\\!\left[-\int_{h_{\mathrm{ch}}}^{y}\frac{\alpha}{T{\scriptstyle(y^{\prime})}}(R_{\sun}+y^{\prime})^{-2}dy^{\prime}\right]}}\quad&&h_{\mathrm{ch}}\leq
y<h_{\mathrm{c}}\\\
&{\textstyle\\!\\!\frac{k_{B}}{N_{A}m_{i}}T_{\mathrm{c}}n_{\mathrm{c}}\exp{\\!\left[-\frac{\alpha}{T_{\mathrm{c}}}\left(\frac{1}{h_{\mathrm{c}}+R_{\sun}}-\frac{1}{y+R_{\sun}}\right)\right]}}&&h_{\mathrm{c}}\leq
y,\par\end{aligned}\end{array}\right.$ (14)

where

${\textstyle
p_{\mathrm{ch}}(y)=\frac{k_{B}}{N_{A}m_{i}}T_{\mathrm{c}}n_{\mathrm{c}}\exp{\left[\int_{h_{\mathrm{ch}}}^{h_{\mathrm{c}}}\frac{\alpha}{T(y^{\prime})}(R_{\sun}+y^{\prime})^{-2}dy^{\prime}\right]}}\,,$

The number density at height $y=h_{\mathrm{c}}$ in the corona is
$n_{\mathrm{c}}=3\times 10^{8}$, $\alpha=\frac{m_{i}GM_{\sun}}{2k_{B}}$, and
$N_{A}$ is the Avogadro number. The internal pressure of the FR is obtained by
proposing a solution close to the equilibrium:

$\displaystyle p_{\text{\tiny{FR}}}(x,y)=p(y)$
$\displaystyle+\tfrac{1}{c}\int_{R}^{r+\frac{\Delta}{2}}B_{\phi}{\scriptstyle(R^{\prime})}j_{z}{\scriptstyle(R^{\prime})}dR^{\prime}$
$\displaystyle-\tfrac{1}{c}\int_{R}^{r+\frac{\Delta}{2}}B_{\text{z}}{\scriptstyle(R^{\prime})}j_{\phi}{\scriptstyle(R^{\prime})}dR^{\prime}.$
(15)

The associated plasma densities are obtained from the adopted equation of
state, i.e.:

${\textstyle\rho=\frac{m_{i}p(y)}{2k_{B}T(y)}}.$ (16)

###  2.3 Setup

Table 1: PSs parameters and class of interaction between the FR and PSs.

|  $B_{0}$[G] |  $B_{\text{PS}}$[G] |  $x_{\text{PS}}$[Mm] |  $y_{\text{PS}}$[Mm] | Class
---|---|---|---|---|---
PS1-L | $1$ | $-1.284$ | $210$ | $-360$ | Ie
PS1-C | $1$ | $-1.284$ | $140$ | $-360$ | Ie
PS1-R | $1$ | $-1.284$ | $70$ | $-360$ | Ie
PS2-L | $2$ | $-2.569$ | $210$ | $-360$ | Ine
PS2-C | $2$ | $-2.569$ | $140$ | $-360$ | Ine
PS2-R | $2$ | $-2.569$ | $70$ | $-360$ | Ie
PS3-L | $0.5$ | $-0.203$ | $210$ | $-180$ | O
PS3-C | $0.5$ | $-0.203$ | $140$ | $-180$ | O
PS3-R | $0.5$ | $-0.203$ | $70$ | $-180$ | O
111Parameter $B_{0}$ determines the magnetic field strength that surrounds the
PS. $B_{\text{PS}}$ modulates the magnetic field strength inside the PS lobes.
The parameters $x_{\text{PS}}$ and $y_{\text{PS}}$ indicate the position of
the dipole that produces the PS lobes. The Class column indicates the initial
scenario (I=inner, O=outer) and if the FR erupts or not (subscripts ‘e’ and
‘ne’, respectively).

Figure 2: Initial scenarios for simulations listed in Table 1. The case name
and the class of interaction are indicated in boxes to the left and right of
each panel, respectively. The colour represents the magnetic energy density,
null points can be noticed in dark violet. The magnetic field lines are drawn
in white.

We perform several simulations to analyse the evolution of a FR interacting
with different PS configurations. For all cases, we establish a single FR
configuration and model the different cases by varying the parameters
describing the magnetic structure of the PS. The simulated FR is warm, its
temperature equals the coronal one ($T_{\mathrm{FR}}=1\leavevmode\nobreak\
\mathrm{MK}$), it has an initial height of $h_{0}=30\leavevmode\nobreak\
\mathrm{Mm}$, a radius of $r=2.5\leavevmode\nobreak\ \mathrm{Mm}$ and a
transition layer thickness of $\Delta=0.25\leavevmode\nobreak\ \mathrm{Mm}$.
Its magnetic parameters are $j_{0}=435\leavevmode\nobreak\ \textrm{statA
cm}^{-2},j_{1}=322\leavevmode\nobreak\ \textrm{statA cm}^{-2}$, $M=1$ and
$d=-3.125\leavevmode\nobreak\ \textrm{Mm}$.

We list in Table 1 the parameters of the selected PSs and, in the last column,
the resulting interaction class with the FR. To fix the height of the null
point ($y_{n}$) we determine the parameter $B_{\textrm{PS}}$ by making zero
equation (11b) ($B_{y,\mathrm{PS}}(x_{\mathrm{PS}},y_{n})=0$). PS1 and PS2
cases correspond to PSs with the null point at a height of $\sim 280$ Mm and
lobes of $\sim 400$ Mm wide. PS3 cases have the null point at height $\sim
140$ Mm and their lobes width is $\sim 200$ Mm. Figure 2 shows the magnetic
energy density and field lines for each case listed in Table 1. The FR is
located to the left of the PS spine. The nomenclature L (left), C (centred)
and R (right) indicates the alignment of the FR respect to the left PS
lobe.These cases are representative of a larger sample of performed
simulations. They cover the following combination of parameters: background
magnetic field $B_{0}=\\{0.5,1,2\\}$ G; null point height
$y_{n}=\\{140,280\\}$ Mm; lobe width $w\sim\\{y_{n},1.5y_{n}\\}$ and FR
alignment $\\{$R, C, L$\\}$. The coupling of the PS and FR magnetic fields
alters the PS shape, resulting in a displacement of the PS null point and the
appearance of a new null point for all cases. We refer to the PS null point,
which is located at a certain height along the spine, as global null point
(GNP). Likewise, we name the new null point, produced by the addition of the
FR, local null point (LNP).

Two initial scenarios are possible when the PS and FR magnetic field are
combined. In one scenario, the LNP is closer to the FR and it forms inside the
PS structure. The magnetic field lines of the PS lobe that overlay the FR form
a confining cage. In the other scenario, the FR field is strong enough to
change the PS topology, bending the left lobe field, and the LNP is associated
with a new spine-like structure outside the PS. Therefore, there is no arcade
over the FR forming a magnetic cage. We refer to the interaction resulting
from the first scenario as class I (inner; top and middle rows of Fig. 2), and
the second scenario results in a class O (outer) interaction (bottom row of
Fig. 2). Class I cases can also be divided according to whether they are
eruptive or non-eruptive events (see subscripts $e$, for eruptive, and $ne$,
for non-eruptive, in Table 1 and Fig. 2). As we show below, the O class has
only eruptive events, possibly because the rearrangement of the topology
favours the ejection. We focus this work in the description and analysis of
the FR dynamic behaviour and evolution according to the interaction class.

##  3 Results

We present two different analyses: a dynamical and a quantitative one. In the
first one, we analyse and compare how the FR trajectories are influenced by
the presence of the LNP and the GNP. For the second one, we study the forces
that are involved in the dynamics, the unsigned magnetic flux of the magnetic
cage, and how the FRs are affected by these factors.

###  3.1  Dynamic behaviour analysis

In this section we analyse the similarities and differences in the FR
deflection depending on the class (I or O) and on whether the event is
eruptive or non-eruptive. To facilitate the interpretation of the trajectories
we use a new reference frame defined as $x^{\prime}=x-x_{\mathrm{GNP}}$
centred in the GNP of each PS, therefore all the spines are centred at 0
$x^{\prime}$-coordinate. All simulations are analysed until the flux rope
reaches a height of $y=600\,\mathrm{Mm}$ or $t=4000\,\mathrm{s}$, whichever
comes first.

####  3.1.1 Class I

Eruptive cases

Figure 3: FR trajectories for the class Ie cases. Dashed lines represent the
path followed by the FR, stars the LNP location, and circles the GNP position
for each case.

Cases PS1-L, PS1-C, PS1-R and PS2-R belong to class I eruptive cases (see top
row panels and middle-row right panel in Fig. 2). All their LNPs are inside
the PS. For the cases PS1-L, PS1-C and PS1-R we consider the same magnetic
configuration of the PS but different horizontal distances between the PS
spine and the FR (see $x_{\text{PS}}$ in Table 1). These relative distances
determine different $(x,y)$ positions of the null points. Figure 3 shows the
trajectory of the FR for the different eruptive cases. The dashed-line of a
given colour represents the FR trajectory, the stars of the same colour
indicate the location of its LNP, and the circles indicate the position of its
GNP.

There is a common behaviour for class I eruptions. Initially, the FR moves
towards the LNP, therefore the location of the LNP determines the direction of
the initial deflection. After that, the LNP is deformed by the displacement of
the FR, and the latter continues to rise towards the new direction of low
magnetic energy. The FR is guided towards the PS spine, located above the GNP.
The final directions of all class Ie trajectories eventually converge to a
path parallel to the PS spine (which is not always radial). The arrival time
and speed at this path will depend on the previous trajectory induced by the
LNP. For example, FRs whose initial trajectory is more aligned with the
direction of the GNP will eject faster (e.g. PS1-C) than those that are
deviated by the LNP in an opposite direction to that of the GNP (e.g. PS1-R).

Non-eruptive cases

Figure 4: FR trajectories for the cases PS1-C (class Ie) and PS2-C (class
Ine). Dashed lines represent the path, stars the LNP location, and circles the
GNP position for each case.

Cases PS2-L and PS2-C are non-eruptive cases of class I (see left and centre
panels in the middle row of Fig. 2). PS2 cases have an almost identical
morphology to PS1 cases (see top and middle rows of Fig. 2). The difference is
that PS2 magnetic fields are twice stronger than PS1 cases. We compare here
PS2-C (non-eruptive) with respect to PS1-C (eruptive); the description is
analogous for PS2-L and PS1-L.

Figure 4 shows the trajectory of PS1-C and PS2-C cases for comparison. The
initial trajectories are remarkably similar, however at some point PS2-C case
starts a decaying phase and consequently no eruption occurs. The FR continues
its descendent motion towards the initial position of the GNP. To understand
this behaviour we display in Figure 5 both cases at time $t=1400$ s. The dark
region highlights the FR location and the colour scale indicates the strength
of the magnetic field lines. It can be seen that the volume of the FR for
PS1-C case is larger than for PS2-C. Studying the environment of the FR, we
note that the magnetic cage, formed by the set of magnetic field lines from
the PS lobe that overlay the FR and confine it, is wider and stronger for
PS2-C. The differences between the two cages are most evident in the upper
section of the cage, where the higher number of lines constituting the PS2-C
cage is clear and their strengths can be compared by the colour levels. In
addition, we observe a marked difference in the response of both cages to the
FRs rise. In case PS1-C, we observe that the cage adapts and follows the shape
of the FR, adopting a lock-like shape. In contrast, the PS2-C cage is not
sufficiently prone to deformation, and this produces a noticeable imbalance
between the fields supporting the FR below and those confining it above. Then,
it is reasonable to infer that this produces a strong magnetic pressure
gradient pushing the FR towards the base of the corona, which could trigger
the decaying phase. The rigidity of the PS2-C cage may also be the reason for
the reduced expansion of the FR, as can be noticed in the animation of the
right panel of Fig. 5 available in the HTML version. The animation shows the
FR rising (until $t=2000\,$s) and its subsequent descent. At this last stage,
the FR suffers a draining that results in the formation of detached magnetic
islands around the FR boundaries. A similar behaviour is observed when
comparing PS1-L and PS2-L cases, which share the same path until PS2-L slows
its upward motion and finally starts the decaying phase. PS2-L also barely
expands and suffers from mass draining.

Summarising, the general behaviour of Class I non-eruptive events is
characterised by a rising and a decaying phase. During the rising phase, the
FR trajectories present the same behaviour than the Class I eruptive events
described above. The magnetic cages of non-eruptive cases withstand the upward
motion and slow down the FR until the decaying phase begins. During the
decaying phase, the FR no longer resists the action of gravity and is guided
by the ambient magnetic field lines towards the chromosphere. In addition,
non-eruptive FRs expand weakly under the pressure of the ambient magnetic
field and they become smaller as their outer parts split into detached
magnetic islands, sometimes completely destroying the identity of the FR.

Figure 5: FR position (dark region) and magnetic field lines strength of PS1-C
(eruptive; left panel) and PS2-C (non-eruptive; right panel) cases for
$t=1400$ s. The animated evolution of the right panel is available in the HTML
version.

####  3.1.2 Class O

The triad PS3-L, PS3-C and PS3-R belongs to the class O events. They have a
common PS structure, but the FR horizontal position is different for each of
them. Because these cases are topologically different from the class I cases
(see bottom row of Fig. 2), the FR trajectories are not affected in the same
way by the null points. This triad has their own spine-like structure whose
base is located at the LNP. Figure 6 shows the trajectories for the PS3 cases.
It can be seen that PS3-L and PS3-C are barely affected by the GNP, heading
initially towards their LNP and then continuing upwards into their own spine
zone, guided by the open magnetic field lines of this spine. However, PS3-R
case, whose initial position is almost equidistant from both null points,
travels between them before reaching the spine of the LNP.

Summarising, the FR trajectories for class O are initially headed towards
their LNP and then continue upwards toward their own spine-like zone. The
events are not influenced by the GNP except when the FR is relatively close to
it. Moreover, simulations in this scenario always erupt, seemingly because the
LNP is directly connected to the open field lines and there is no magnetic
cage above the FR.

Figure 6: FR trajectories for the class O cases: PS3-R, PS3-C and PS3-L.
Dashed lines represent the path, stars the LNP location, and circles the GNP
position for each case.

###  3.2 Quantitative analysis

Figure 7: Magnetic field lines for events PS1-C (left) and PS2-C (right). The
shaded light blue areas represent the magnetic cage above the FR, magenta and
indigo dots indicate the initial position for PS1-C and PS2-C, respectively.
The solid line represents the internal trajectory of the FR through which the
flux of the magnetic cage is quantified. Figure 8: Total unsigned magnetic
flux per unit length of the magnetic cage for PS1 and PS2 cases.

In this section we will focus on the magnetic cage and its effect on FR
evolution. To be precise, we define a magnetic cage as the structure formed by
all the field lines enclosing the FR and whose both ends are attached to the
base of the corona (height $y=0$). As suggested previously by Fig. 5, the
magnetic cage of the non-eruptive events is larger and more intense than that
of the eruptive cases. Figure 7 shows the magnetic cages for the PS1-C and
PS2-C cases shaded in light blue. Inspired by the results of Li et al., (2020,
2021, 2022), we also determine the total unsigned magnetic flux to quantify
the strength of the magnetic cages. Taking advantage of the symmetry
considered in the $z$-direction, we calculate the initial magnetic flux per
unit length $\phi_{B}$ through a path outlined by the FR trajectory as
follows:

$\phi_{B}=\tfrac{1}{L_{z}}\int_{A}|\@vec{B}\cdot\@vec{dA}|=\int_{\gamma}|B_{\perp}|\,dS,$
(17)

where $B_{\perp}$ is the magnetic field transverse to a curve $\gamma$ defined
by the FR path (denoted by the solid coloured lines in Fig. 7). Figure 8 shows
the total unsigned magnetic flux of each magnetic cage for all PS1 and PS2
cases (PS3 cases do not produce magnetic cages). Note that the magnetic flux
values for the non-eruptive cases (PS2-L and PS2-C) are remarkably large in
comparison to the eruptive cases.

Figure 9: Parameter map for the simulation set. The colour bar represents the
magnetic cage flux strength. $B_{0}$ parameter is the background magnetic
field related to the PS, $w_{1}$ and $w_{2}$ refer to the PS lobe widths,
R-C-L are the FR alignments, and $y_{n}$ is the PS null point height. For each
box the dynamical behaviour is indicated.

To understand how the dynamical behaviour is affected by the simulation
parameters, we include in Figure 9 the magnetic cage flux (represented by the
colours of the colour bar) for the whole simulation set, as a function of
$B_{0}$ ($y$-axis) and width-alignment ($x$-axis). The widths ($w$) are
$w_{1}\sim y_{n}$ and $w_{2}\sim 1.5y_{n}$, being $y_{n}$ the height of the
null point, and the alignment R-C-L as described in Section 2.3. We also
separate the cases according to $y_{n}$ (top and bottom parts of the plot)
with the values denoted on the right. For each case we indicate its
classification as we defined in previous section. We notice again the
correlation between larger magnetic cage fluxes and non-eruptive cases. In
addition, these cases are more related with stronger $B_{0}$ magnetic fields
and narrower pseudostreamers ($w_{1}$). This is expected as these parameters
influence the magnetic flux of the pseudostreamer lobe. However, the magnetic
cage flux will also depend on the position and parameters of the FR, i.e. on
how many lobe lines actually belong to the cage.

Figure 10: Vertical velocity $v_{y}$ as a function of time for PS1 and PS2
cases.

We also analyse the evolution of the FR variables for PS1 and PS2 cases to
understand how they are affected by the magnetic cage. PS3 cases are not
included since they do not present a magnetic cage and, moreover, they follow
the trends of the class Ie. First, we analyse the evolution of FR velocity and
total force in the $y$-direction. Figure 10 shows the vertical velocity curves
($v_{y}$) up to $t=1000\leavevmode\nobreak\ $s. The initial force for non-
eruptive cases (PS2-L and PS2-C) is stronger and heads the FRs towards the LNP
with notably higher speeds, likely due to the closer proximity to the LNP.
However, after reaching the maximum value, $v_{y}$ of Ine events decreases
more and steeper than $v_{y}$ of Ie events. Eventually $v_{y}$ becomes
negative and the decaying phase of the FR starts. From the separate analysis
of the force components (magnetic pressure and tension, gas pressure and
gravity, not shown here), we find that the magnetic pressure gradient is the
main responsible for the abrupt deceleration and the descent of the FR, after
which gravity is the dominant decelerating force. This result is in agreement
with the qualitative analysis presented in the previous section, in which we
note that the concentration of field lines over the FR (see Fig. 5) seems to
be responsible for exerting this magnetic pressure force. Gravity becomes the
leading force once the FR is “channelled” by the lobe magnetic field lines
(decaying phase).

Figure 11: Average plasma density of the FR as a function of height. The
dashed grey line represents the plasma density of the corona. The lower panel
shows the corresponding ratio between the plasma density for each case and the
coronal plasma density $\rho_{\mathrm{c}}$.

From the analysis of the FR variables, we note the major differences (between
Ie and Ine cases) in the evolution of the hydrodynamic ones. As we also
mentioned in the previous section, the volume of the non-eruptive FRs remains
small, contained by the strong magnetic cage surrounding them. Thus, the
plasma density and gas pressure for non-eruptive FRs is higher than for the
eruptive FRs, which manage to expand. Figure 11 shows the evolution of the FR
average plasma density as a function of height, together with the coronal
plasma density $\rho_{\mathrm{c}}$ (dashed grey line). Initially, all FRs are
overdense and they quickly decrease their average density since they are out
of external equilibrium. Afterwards, the density continues to decrease as the
FRs expand, following the drop in ambient pressure with altitude. However,
since the non-eruptive FRs (PS2-L and PS2-C) practically stop expanding, their
densities tend asymptotically to a certain value. In the bottom panel of Fig.
11 we present the ratio between the FR and coronal density, which highlights
the balance between the weight and the buoyant force. We note an important
difference between eruptive and non-eruptive cases, while the former manage to
reach densities similar to that of the corona, the non-eruptive ones remain at
more than twice the coronal density due to the lack of expansion.
Consequently, the buoyant force of these last cases is not strong enough to
overcome the gravitational field and the action of the magnetic cage to
produce the eruption.

##  4 Discussion and Conclusions

In this work we analyse the dynamic behaviour of a FR located near an isolated
PS. The magnetic configuration produces the emergence of two magnetic null
points associated with both structures: a LNP (local null point) formed by the
cancellation of the FR and PS magnetic fields, and a GNP (global null point)
related to the PS itself. We note that the LNP is determinant for the early
evolution of the FR. All simulated cases show an initial deflection due to the
attraction towards this point of low magnetic energy. The subsequent evolution
depends on whether the FR is enclosed by the PS lobes (class I events) or not
(class O), showing that the hierarchy of the null points depends on the
topology. In class O, the LNP is associated to an intrinsic spine-like
configuration and the FR is guided by its open magnetic field lines instead of
travelling towards the PS spine. For class I events, a second deflection can
take place by the influence of the GNP, directing the FR to the PS spine. In
this scenario, it is possible that the eruption fails. We determine that the
magnetic cage, formed by the magnetic field lines from the PS lobe that
encloses the FR, plays a crucial role in curbing the eruption.

The non-eruptive cases, which initially reach higher velocities, are quickly
decelerated by the magnetic cage. The cage field lines are compressed instead
of adjusting to the rise of the FR, producing high magnetic pressure gradients
that impulse the FR back to the surface. Also, we note for these cases that
the expansion of the FR is inhibited by the magnetic cage, keeping it
overdense and less buoyant, which helps to prevent the eruption. Thus, we
quantified the total unsigned magnetic fluxes of the cages, obtaining that in
the non-eruptive cases the average value is almost six times higher than in
the eruptive cases. This magnitude can be interpreted as a measure of the
magnetic cage resistance. We also showed that cases with stronger magnetic
field $B_{0}$ and narrower PS lobes are prone to be non-eruptive.

We show that the combination of a FR with a PS magnetic structure is
topologically complex. Although the relative position between the FR and PS
centre plays an essential role in predicting the non-radial motions of the FR
trajectory, the magnetic flux contained in the magnetic cage seems to be the
key parameter in determining whether an eruption can occur or not, in
agreement with previous studies. Hence, we consider of utmost importance to
attain improved magnetic field measurements such as those to be provided by
missions like Solar Orbiter, PUNCH (Polarimeter to UNify the Corona and
Heliosphere), and Aditya, among others, in order to analyse more observational
events that can be compared with our results and to refine numerical models
that contribute to space weather forecasts.

###### Acknowledgements.

AS is doctoral fellow of CONICET. MC, GK, HC and AC are members of the Carrera
del Investigador Científico (CONICET). AS and MC acknowledge support from
ANPCyT under grant number PICT No. 2016-2480. AS, MC, MVS and AC also
acknowledge support by SECYT-UNC grant number PC No. 33620180101147CB. AS, MC,
MVS and AC acknowledge support from PIP under grant number No.
11220200103150CO. MVS acknowledges support from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 724326). HC appreciates support from grants
MSTCAME8181TC (UTN) and PIP 11220200102710CO (CONICET). Also, we thank the
Centro de Cómputo de Alto Desempeño (UNC), where the simulations were carried
out.

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2022

[1]Asuka Inoue [2,3]Akira Furusawa

1]NTT Device Technology Labs, NTT Corporation, 3-1, Morinosato Wakamiya,
Atsugi, 243-0198, Kanagawa, Japan

2]Department of Applied Physics, School of Engineering, The University of
Tokyo, 7-3-1, Hongo, Bunkyo, 113-8656, Tokyo, Japan

3]Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing,
2-1, Hirosawa, Wako, 351-0198, Saitama, Japan

# 43-GHz bandwidth real-time amplitude measurement of 5-dB squeezed light
using modularized optical parametric amplifier with 5G technology

<EMAIL_ADDRESS>Takahiro Kashiwazaki
<EMAIL_ADDRESS>Taichi Yamashima
<EMAIL_ADDRESS>Naoto Takanashi
<EMAIL_ADDRESS>Takushi Kazama
<EMAIL_ADDRESS>Koji Enbutsu<EMAIL_ADDRESS>Kei Watanabe<EMAIL_ADDRESS>Takeshi Umeki
<EMAIL_ADDRESS>Mamoru Endo<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [

###### Abstract

Continuous-variable optical quantum information processing (CVOQIP), where
quantum information is encoded in a traveling wave of light called a flying
qubit, is a candidate for a practical quantum computer with high clock
frequencies. Homodyne detectors for quadrature-phase amplitude measurements
have been the major factor limiting the clock frequency. Here, we developed a
real-time amplitude measurement method using a modular optical parametric
amplifier (OPA) and a broadband balanced photodiode that is commercially used
for coherent wavelength-division multiplexing telecommunication of the fifth-
generation mobile communication systems (5G). The OPA amplifies one
quadrature-phase component of the quantum-level signal to a loss-tolerant
macroscopic level, and acts as a “magic wand,” which suppresses the loss after
the OPA from 92.4% to only 0.4%. When the method was applied to a broadband
squeezed vacuum with a center wavelength of 1545.32 nm, we observed 5.2 $\pm$
0.5 dB of squeezing from DC to 43 GHz without any loss correction. The
marriage of CVOQIP and 5G technology arranged by the modular OPA will lead to
a paradigm shift from the conventional method of using stationary qubits,
where the information is encoded in a standing wave system, to a method using
flying qubits for ultra-fast practical quantum computation. This means that
quantum computer research will move from the stage of developing machines that
execute only specific quantum algorithms to a stage of developing machines
that can outperform classical computers in running any algorithm.

###### keywords:

Optical quantum computer, Homodyne detection, Optical parametric amplifier

##  1 Introduction

Figure 1: (a) Bandwidth (BW) relationship between squeezed vacuum and homodyne
detector. $f_{\textrm{c}}$, center frequency of the squeezed
vacuum;$f_{\textrm{HD}}$, cut-off frequency of the homodyne detector; LO,
local oscillator. (b) Concept of a multi-core quantum processor. $\Delta f$,
frequency spacing. (c) Quadrature-phase amplitudes $X_{f}(t),P_{f}(t)$ of the
electromagnetic filed of light to be measured with the temporal mode function
$f(t)$. (d-f) Three kinds of homodyne apparatus measuring the $X$ component
($\theta=0$). Each $I_{\textrm{HD}}(t,0)$ shows the output signal. The output
signals are affected by the detection efficiencies and the bandwidths. (d)
Standard homodyne system for quantum light detection. The photodiodes used in
the homodyne detector have almost unity detection efficiencies, but the system
bandwidth $f_{\textrm{HD}}$ is typically less than a GHz. (e) Homodyne system
with broadband balanced detector, which is often used in 5G applications. The
bandwidth $f_{\textrm{HD}}$ is beyond a GHz but the system efficiency is not
high (typically $\sim$50%). (f) Proposed homodyne system with an OPA and a
balanced detector for 5G applications. The $X$ component of the input light
field is amplified by the OPA excited by the pump light and is detected by the
homodyne detector with the 5G balanced detector.

Research on continuous-variable optical quantum information processing
(CVOQIP), in which quantum information is encoded in an optical
electromagnetic field, is aimed at realizing a fault-tolerant universal
quantum computer Lloyd1999 ; OBrien2009 ; Takeda2019 ; Fukui2022 ;
Asavanant2022 . This form of processing has a high carrier frequency, which
can raise the upper limit of the clock frequency. Moreover, its decoherence is
small even at room temperature and atmospheric pressure because the photon
energy is high compare to the environment.

In CVOQIP, measurement-based quantum computation (MBQC) is a reasonable way of
performing large-scale quantum computations Menicucci2011a ; Ukai2015 , rather
than circuit-model computations. MBQC uses a cluster state as a resource
state, which are special multipartite entangled states. The cluster state
contains a superposition of all input-output relations in quantum computation,
and can be regarded as superposition of quantum lookup tables. In order to
select a desired input-output relation from them, that is, to perform the
desired computation, we only need to perform local measurements for each qubit
by collapsing the wavefunction. It is also called a one-way quantum
computation because the measurements perform a quantum calculation and
collapse the wavefunction simultaneously. It was initially proposed for a
standing wave system, that is, stationary qubit Raussendorf2001 . In this
case, the same qubit must be kept until the calculation is completed, so the
lifetime of the qubit, or its memory time, must be longer than the calculation
time.

On the other hand, in the case of CVOQIP, the wave packets from the cluster
state resource are treated as flying qubits Lloyd1999 . Since homodyne
detectors immediately measure each wave packet, the qubit is consumed, so
there is no need to consider the lifetime Asavanant2021 ; Larsen2021 . In
other words, it can be called a one-pad quantum computation (OPQC).

The processor of an optical OPQC consists of a time-domain multiplexed (TDM)
cluster state and a homodyne measurement that measures the quadrature-phase
amplitude of each wave packet. The TDM cluster state is created on the basis
of a continuous-wave (CW) squeezed vacuum Yokoyama2013 ; Yoshikawa2016 ;
Asavanant2019 ; Larsen2019 . In a squeezed vacuum, the noise of the
electromagnetic field at a particular quadrature-phase component is below the
vacuum. A squeezed vacuum is generated by creating quantum correlations
between photons contained within a given period often through nonlinear
optical processes such as parametric down conversion or four-wave mixing
Andersen2016 . When two squeezed vacua are interfered with a beamsplitter,
quantum entanglement is created between the two spatial modes at the output.
The entanglement is time-delayed by an optical delay line and further passed
through an asymmetric interferometer to extend the entanglement between time-
different wave packets Menicucci2011a ; Ukai2015 .

The broader the squeezing bandwidth is, the shorter the quantum correlation
time, defined by the inverse of the bandwidth of the squeezed vacuum, and the
more wave packets can be “packed” per unit time. In other words, the upper
limit of the clock frequency is the bandwidth of the squeezed vacuum. It has
been reported that low-loss periodically poled lithium niobate (PPLN)
waveguides can generate 6-dB of squeezing in a 6-THz bandwidth with a center
wavelength of 1545 nm Kashiwazaki2021  and 4-dB squeezing in a 25-THz
bandwidth with a center wavelength of 2 $\mu$m Nehra2022 . That is, a THz
clock frequency can be realized in principle.

However, the bandwidth of the homodyne detectors $f_{\textrm{HD}}$ and other
devices is currently limited to about GHz, so the wide bandwidth of the
squeezed vacuum is wasted, as depicted in Fig. 1 (a). To measure quantum
light, the detection efficiency of a photodiode must be very high, but a
photodiode specifically designed for this purpose will inevitably have a
narrow bandwidth. Realistically, the bandwidth is limited to a few hundred MHz
Mehmet2011 ; Huang2013a ; Vahlbruch2016a  or at best a few GHz with state-of-
the-art silicon photonics technology Tasker2020 ; Bruynsteen2021 .

Recently, a technique employing an optical parametric amplifier (OPA) has been
used to measure a broadband squeezed vacuum Shaked2018 ; Takanashi2020 ;
Nehra2022 . OPAs act as phase-sensitive optical amplifiers that can amplify
one quadrature-phase component without adding noise, i.e., with a noise figure
(NF) of 0 dB in principle Caves1982 ; Umeki2011 ; Kazama2021 . This is a
significant difference from phase-insensitive optical amplifiers (e.g.,
erbium-doped fiber amplifiers, Raman amplifiers, semiconductor optical
amplifiers, etc.), which cannot cross the NF = 3 dB barrier Caves1982 . A
quantum-level signal can be converted into a macroscopic level with a low-loss
OPA with a sufficiently high gain acting as an optical pre-amplifier for
quantum measurements Manceau2017 . Consequently, optical losses and electrical
noise are suppressed, which significantly increase the degrees of freedom of
the measuring instrument. However, the previous studies made only narrow
bandwidth measurements or power measurements like spectrum analyzers
Shaked2018 ; Takanashi2020 ; Nehra2022 , while OPQC requires real-time
broadband quadrature-phase amplitude measurements made by homodyne detectors.

In this paper, an OPA amplified one quadrature-phase component of the light to
be measured, then we detected its output with a balanced photodiode and a
broadband amplifier. The OPA acts as an optical pre-amplifier and suppresses
the losses after the OPA from 92.4% to only 0.4%, yielding an overall homodyne
measurement system efficiency of 79%. We measured a broadband squeezed vacuum
with a center frequency of 194.0 THz (wavelength of 1545.32 nm) generated by
another OPA Kashiwazaki2021 . We observed 5.2-dB squeezing in the band from
near DC to 43 GHz without any loss correction. Thus this method can overcome
the loss problem, which is the main concern in the field of silicon-photonics
based quantum information processing.

The critical component of this work is a fiber-coupled, low-loss and high-gain
broadband OPA with a PPLN waveguide. Many of the other components are commonly
used in digital coherent telecommunications in the fifth-generation mobile
communication systems (5G). We believe that the resulting marriage of CVOQIP
and 5G technology will lead directly to an optical quantum computer with a
clock frequency exceeding 40 GHz. The 5G elements used in our demonstration
were the balanced photodiodes and an electrical amplifier. In addition, as
illustrated in Fig. 1 (b), dividing the bandwidth of the squeezed vacuum using
a dense wavelength division multiplexing (DWDM) coupler, as is done in WDM
digital coherent communications, enables multiple independent quantum
processors to be realized from a single light source. In this case, optical
quantum computer can be realized by applying a high-repetition-rate optical
frequency comb to the phase-coherent local oscillator light for each homodyne
detector.

##  2 Results

###  2.1 Homodyne detection with OPA

The electromagnetic field of light with a carrier frequency of $\omega_{0}$ is
represented by using annihilation and creation operators, $\hat{A}(t)$ and
$\hat{A}^{\dagger}(t)$, excluding the oscillation term associated with the
carrier frequency, as follows:

$\displaystyle\hat{E}(z,t)$
$\displaystyle\propto\hat{A}(t)e^{i(kz-\omega_{0}t)}+\hat{A}^{\dagger}(t)e^{-i(kz-\omega_{0}t)}$
(1)
$\displaystyle=\sqrt{2}\left(\hat{X}(t)\cos(kz-\omega_{0}t)+\hat{P}(t)\sin(kz-\omega_{0}t)\right).$
(2)

Here, these operators satisfy the commutation relation
$[\hat{A}(t),\hat{A}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$, where
$\delta(t)$ is the Kronecker delta function, and
$\hat{X}(t)\equiv\frac{\hat{A}(t)+\hat{A}^{\dagger}(t)}{\sqrt{2}},\hat{P}(t)\equiv\frac{\hat{A}(t)-\hat{A}^{\dagger}(t)}{\sqrt{2}i}$
are called the quadrature-phase amplitude operators. We often consider
wavepackets of light defined by a temporal mode function $f(t)$ as
$\hat{A}_{f}(t)\equiv\int dt^{\prime}f(t^{\prime}-t)\hat{A}(t^{\prime})$, and
the quadrature-phase amplitude operators of this mode are defined as
$\hat{X}_{f}(t)\equiv\frac{\hat{A}_{f}(t)+\hat{A}_{f}^{\dagger}(t)}{\sqrt{2}},\hat{P}_{f}(t)\equiv\frac{\hat{A}_{f}(t)-\hat{A}_{f}^{\dagger}(t)}{\sqrt{2}i}$.
Figure 1 (c) shows examples of measurement results which correspond to
eigenvalues $X_{f}(t)$ and $P_{f}(t)$ of $\hat{X}_{f}(t)$ and
$\hat{P}_{f}(t)$, respectively.

When we perform balanced homodyne detection using local oscillator (LO) light
with an optical frequency of $\omega_{0}$ as shown in Fig. 1 (d) and (e), the
output can be written as

$\displaystyle\hat{I}_{\textrm{HD}}(t,\theta)\propto\hat{X}_{\textrm{HD}}(t)\cos\theta+\hat{P}_{\textrm{HD}}(t)\sin\theta$
(3) $\displaystyle\hat{X}_{\textrm{HD}}(t)\equiv\int
dt^{\prime}f_{\textrm{HD}}(t^{\prime}-t)\hat{X}_{f}(t^{\prime})$ (4)
$\displaystyle\hat{P}_{\textrm{HD}}(t)\equiv\int
dt^{\prime}f_{\textrm{HD}}(t^{\prime}-t)\hat{P}_{f}(t^{\prime}),$ (5)

where $\theta$ is the phase between the signal and LO lights, and it is
assumed that the LO light is sufficiently strong and the quantum fluctuations
are negligible. $f_{\textrm{HD}}$ is the temporal mode function of the
instruments, including the homodyne detector, amplifiers, filters,
oscilloscope and so on. The Fourier transform of $f_{\textrm{HD}}(t)$ is
related to the frequency response of the measurement system. For CVOQIP
fields, the time-domain data are used not only to read the result but also to
process the signal by using feedforward operations.

For example, a homodyne detector with high detection efficiency
($\eta_{\textrm{HD}}\sim 1$) is typically used in the case of a standard
homodyne system for measuring the quantum state of light (Fig. 1 (d)). Several
research groups have used this kind of system to measure high-level squeezed
vacua Mehmet2011 ; Huang2013a ; Vahlbruch2016a  or non-Gaussian states with
strong Wigner negativity, such as Schrödinger cats Neergaard-Nielsen2006 ;
Gerrits2010 ; Asavanant2017 ; Takase2022  or photon-number states Neergaard-
Nielsen2007a ; Yukawa2013 . However, such a homodyne system is specialized for
achieving high-detection efficiency, and the bandwidth is not very wide
(typically less than GHz order).

In contrast, broadband balanced detectors lift the bandwidth beyond a GHz or
even higher. Recently, GHz homodyne detectors built with silicon photonics
technology Tasker2020 ; Bruynsteen2021  have been demonstrated; one of them
measured a broadband squeezed vacuum Tasker2020 . Nevertheless, to increase
the bandwidth, it is necessary to use photodiodes with a thin active layer and
a small photosensitive area, which is not suitable for achieving high
detection efficiency. Although the coupling efficiency can be increased by
integration, there are still limitations. Losses and electrical noises
directly degrade the signal-to-noise ratio (SNR) of the homodyne system, which
is critical for CVOQIP applications (Fig. 1 (e)).

Here, we amplify the $X$ component by using an OPA and detect the output with
a homodyne detector, as depicted in Fig. 1 (f). The output signal is
represented as

$\displaystyle\hat{I}_{\textrm{HD,OPA}}(t,\theta)\propto\sqrt{G}\hat{X}_{\textrm{HD}}(t)\cos\theta+\frac{1}{\sqrt{G}}\hat{P}_{\textrm{HD}}(t)\sin\theta,$
(6)

where $G$ is the gain of the OPA. Instead of losing information in the $P$
component, the $X$ component is amplified, as shown in Fig. 1 (f). By setting
$\theta=0$, the $X$ component of the quadrature-phase amplitude can be
measured. The ideal OPA acts as a phase-sensitive optical amplifier and does
not add noise when amplifying the $X$ component, thus suppressing the loss of
the broadband homodyne detector.

Let the efficiencies of the homodyne detector and OPA be $\eta_{\textrm{HD}}$
and $\eta_{\textrm{OPA}}$, respectively. The effective efficiency of the
system can be written as

$\displaystyle\eta_{\textrm{eff}}=\frac{\eta_{\textrm{OPA}}\eta_{\textrm{HD}}}{\eta_{\textrm{HD}}+\frac{1-\eta_{\textrm{HD}}}{G}}.$
(7)

From this expression, if the loss of the OPA is small compared with the loss
of the homodyne detector and the gain of the OPA is sufficiently high, $G\gg
1$, the OPA can suppress for the loss of the homodyne detector (see Methods).
Optical losses, the detection efficiency of a broadband balanced photodiode,
and electrical noise in the broadband measurement equipment would easily
increase the total loss of the homodyne system to more than 90%, corresponds
to $\eta_{\textrm{HD}}=10\%$. Consider, for example, a case where the loss is
90% ($\eta_{\textrm{HD}}=10\%$): if the OPA has a gain $G$ of 35 dB and an
efficiency $\eta_{\textrm{OPA}}$ of 79% (corresponds to 1-dB NF), the loss
after the OPA can be improved from 90% to only 0.3%. Moreover, further
improvements in efficiency could be achieved in the future if a low-loss and
high-gain OPA could be realized.

###  2.2 Experimental apparatus

Figure 2: (a) Experimental apparatus of the real-time quadrature phase
amplitude measurement using the proposed homodyne system. OPA1, optical
parametric amplifier used for generating squeezed vacuum; OPA2, optical
parametric amplifier for amplifying the$X$ component of the light to be
measured; 10:90 and 50:50, fiber-based beamsplitters with coupling ratios of
90% and 50%, respectively; LO, local oscillator light (1545 nm); Pump, pump
light for the OPAs (773 nm); VOAs, variable optical attenuators; DLs, optical
delay lines; BD, balanced photodiode with a bandwidth of 43 GHz; AMP,
broadband amplifier; SPL, broadband power splitter; OSA, optical spectrum
analyzer; ESA, electrical spectrum analyzer, FBC1, feedback circuit for
locking the phase among the squeezed vacuum and the pump light at OPA2; FBC2,
feedback circuit for locking the phase at the amplified signal and LO. (b)
Output signals measured by a real-time oscilloscope. The black trace shows the
electrical noise floor when all lights are blocked. The red trace shows the
shot noise of the LO when the photocurrent of a single photodiode is 3.0 mA.
(c) Frequency responses of several photocurrents calculated by fast Fourier
transformation of the oscilloscope data.

Figure 2 (a) shows the experimental apparatus of our homodyne system. The
fundamental light is from a continuous-wave fiber laser (NKT photonics, X15)
at 1545.32 nm (not shown). The light to be measured is a broadband squeezed
vacuum generated by a fiber pig-tailed OPA (OPA1 in the figure)
Kashiwazaki2021  . The output fiber of OPA1 is connected to the input fiber of
OPA2 (used as a phase-sensitive optical amplifier), which is excited by pump
light with a wavelength of 773 nm. 10% of the OPA2 output is used for
monitoring the output spectrum by an optical spectrum analyzer (OSA, Yokogawa
AQ6370D), and locking the phase between the squeezed vacuum and the sub
harmonic of the pump light by a feedback circuit (FBC1) Takanashi2020  (see
the supplementary information). The remaining 90% is sent to a 50:50 fiber
beamsplitter, where it interferes with the local oscillator (LO) light and is
received by a balanced photodiode (BD, $\textrm{u}^{2}\textrm{t}$, BPDV21x0R).
Delay lines (DLs) and variable optical attenuators (VOAs) are inserted in each
path to adjust the optical path length and power balance. The two photodiodes
are reverse biased by biasing circuits, and each photocurrent
$I_{\textrm{PD}}$ is monitored (not shown). The output of the balanced
photodiode is amplified by a broadband amplifier (AMP, SHF, S807 B, 55-GHz
bandwidth). The output is split by a broadband power splitter (SPL, Anritsu,
W241A); then, one part is measured by a real-time oscilloscope (Keysight,
DSO-Z 634A, 63-GHz bandwidth). The other part is monitored by an electrical
spectrum analyzer (ESA, Keysight, EXA N9010B). The ESA is also used to
generate an error signal to lock the phase between the LO light and the output
of OPA2 by a feedback circuit (FBC 2) (see the supplementary information for
details).

###  2.3 Characterization of the homodyne system

In the following experiments, the pump power of OPA2 was set to 1.2 W. The
parametric gain $G$ and NF of OPA2 were 35 dB and 1 dB, respectively (see the
supplementary information and the reference Kazama2021 ). The measured loss
after the OPA2 was 92.4%, then from Eq.7 the effective efficiency of the total
homodyne system is 79%. Figure 2 (b) shows the raw data of the homodyne
detector. The black trace shows the electrical noise floor when all the light
is blocked. The red trace shows the shot noise with the LO when the
photocurrent of each photodiode is set to 3.0 mA. We acquired the output
signal for 78.2 ns. Figure 2 (c) shows fast Fourier transform (FFT) traces of
the acquired data with different LO powers. These traces were calculated by
averaging the results of 8192 measurements taken over a 0.8-ms period and all
the frequency response traces are averaged for 8192 traces. The traces show a
flat frequency response up to 10 GHz. The 3-dB bandwidth of our system is 43
GHz of the BD, and even at that frequency, 20 dB of SNR is obtained when the
photocurrent is 3.0 mA (red trace). All the traces are fall off beyond the
frequency of 63 GHz, which is the the edge of the oscilloscope’s bandwidth.
Note that OPA2 has a THz bandwidth, which means a much wider bandwidth can be
obtained if faster electronics can be used.

###  2.4 Measurement of broadband squeezed light

Figure 3: Measurement results of the broadband squeezed vacuum. (a) Raw data
acquired by the oscilloscope. Each point represents the quadrature phase
component. (left) Anti-squeezing. (center) Shot noise. (right) Squeezing. (b)
Histograms calculated from (a). (d) Frequency domain data calculated by FFT
from the time-domain data. Each trace is averaged for 8192 frames of 12512
points of data. The pump power for OPA1 is 438 mW. (d) Relative squeezing and
anti-squeezing levels in the frequency domain for several pump powers.
(e)Pump-power dependence of the squeezing (red circles) and anti-squeezing
(blue circles) levels. The dashed lines show fitting results. (f) Calculated
(lines) and measured (markers) squeezing levels with additional loss after
OPA2. The gain of OPA2 was set to 0 dB (dashed lines) and 35 dB (solid lines).

Figure 3 (a) shows the raw data corresponding to quadrature-phase amplitudes
of the squeezed light of the anti-squeezing and squeezing components (left and
right, respectively). The center shows the shot noise when the pump for OPA1
is set to 0 mW. Figure 3 (b) is a histogram based on the measurement results
of Fig. 3 (a). Figure 3 (a) and (b) clearly show quadrature squeezing of the
light to be measured.

The FFTs of these data are shown in Fig. 3 (c), and the normalized squeezing
level calculated for several pump powers of OPA1 is shown in Fig. 3 (d). From
these figures, it can be seen that the homodyne system successfully measured a
5.2-dB broadband squeezed vacuum from DC to 43 GHz. Note that the peaks at 34
GHz are measurement artifacts caused by the oscilloscope.

Figure 3 (e) shows the squeezing and anti-squeezing levels as the function of
pump power for OPA1. The occupied circles are calculated from the variance of
time-domain data like in Fig. 3 (a), and the error bars are estimated from
8192 data sets. For a pump power of 438 mW, a squeezing level of $5.2\pm 0.5$
dB and an anti-squeezing level of $13.9\pm 0.1$ dB are obtained. The dashed
line is the fitting to the function:

$\displaystyle R_{\pm}(P)=L+(1-L)\exp\left(\pm 2\sqrt{aP}\right)$ (8)

where $\pm$ represents anti-squeezing (+, blue) or squeezing (-, red), $L$
represents the total effective loss of the system, including of OPA1 and the
homodyne system, and $a$ is a second-harmonic-generation coefficient in the
unit of $\textrm{W}^{-1}$. The fitting result shows the system loss of 29%,
including losses of OPA1, OPA2, optical propagation, detector, and the
electrical noise of the instruments. This value limits the squeezing level
that can be measured with this apparatus to $5.2$ dB.

In order to confirm the performance of the suppressing losses after OPA2, we
inserted a VOA (not shown in Fig. 2 (a)) after OPA2 to worsen
$\eta_{\textrm{HD}}$ and measured the squeezing level as a function of the
loss as shown in Fig. 3 (f). Note that the measurements in Fig. 3 (f) are not
broadband measurements by the oscilloscope but rather narrowband measurements
made with ESA at a sideband frequency of 100 MHz and we did not lock the LO
phase (see supplementary information). The dashed and solid lines are
calculated traces when $G=0$ and $35$ dB. The markers with error bars
represent the measured values. It is clear from this figure that OPA2 with
$G=35$ dB effectively suppresses the loss after OPA2 from 92.4% to 0.4%, and
the system can measure the original squeezing level.

##  3 Discussion

The most significant advantage of optical quantum computers is that the upper
limit of the clock frequency is by far the highest compared with other
methods. The bandwidth of the squeezed light used to generate the cluster
states is above THz, meaning that a THz-clocked quantum computer can be
realized in principle. However, the bandwidth on the measurement instruments,
including the homodyne detectors, has not kept pace, and a large part of the
bandwidth of the squeezed light is wasted. We have experimentally demonstrated
a high-detection efficiency (79%) and wide-bandwidth (43 GHz) homodyne system
using an OPA module and commercially available 5G components. The OPA
drastically suppress the losses after OPA2 from 92.4% to only 0.4%. With this
system, we performed real-time measurements of the quadrature-phase amplitude
of broadband squeezed vacuum, obtaining a squeezing level of 5.2$\pm$0.5 dB
without loss correction. This value exceeds one of the thresholds (4.5 dB) for
cluster state generation Asavanant2019 , indicating that this method can be
applied to quantum calculations. In fact, except for the handling of non-
Gaussian quantum states Gottesman2001 ; Fukui2022 , the necessary elements for
optical quantum computation are LO phase switching and feedforward systems,
which are parts of classical information processing. Even at present, there
are electro-optical modulators and electronics above 40 GHz in bandwidth as 5G
technology Wang2018a ; Pittala2022 , and when “Beyond 5G” or 6G technologies
Dang2020  appear in the future, the speed of classical information processing
will be further increased. This means that the broadband homodyne detector
developed in this study will make it feasible for the clock frequency of
optical quantum computers to exceed 40 GHz.

We conclude this paper by proposing a multi-core optical quantum processor by
effectively using the full-bandwidth of squeezed vacuum with our homodyne
system.

Quantum entanglement between two frequency sidebands is the essential feature
of a squeezed vacuum. In this study, we measured the quantum correlation in
the $\pm$43 GHz bandwidth around a frequency $f_{\textrm{c}}$ of 194.0 THz, as
shown in Fig. 1 (a). Therefore, by dividing the optical spectrum into
frequency-band pairs with DWDM couplers (e.g., with a frequency spacing
$\Delta f=100$ GHz) and applying our homodyne system for each as illustrated
in Fig. 1 (b), a multi-core optical quantum processor can be realized, wherein
the operating clock of each processor exceeds 40 GHz. As well, optical
frequency comb technology can be used to prepare phase-coherent LO beams for
each homodyne system Yang2021 ; Corcoran2020 ; Chang2022 . This research shows
that CVOQIP is highly compatible with mature 5G technology.

Increasing the clock frequency and core multiplexing technologies have
dramatically enhanced the processing speed of classical computers, which
created the current information society. The method proposed in this paper
shows that high-clock frequency and multi cores are also possible in optical
quantum computers. Ultra-fast multi-core optical quantum processors will
catapult quantum computer research into a new era, where versatile quantum
computers surpass classical computers in any algorithm, not just in specific
quantum algorithms.

##  4 Methods

###  4.1 OPAs

We used two kinds of OPA. Both are 45-mm-long, ZnO-doped PPLN waveguides, but
their fabrication processes and waveguide structures are different. This is
because their different applications require modules with different
characteristics. OPA1, which is used for generating the broadband squeezed
vacuum, is fabricated by mechanically polishing and the thickness and the
width of the waveguide are about 8 $\mu$m and 8.6 $\mu$m. Thanks to the smooth
surface and relatively large waveguide core, the waveguide loss is only 7% and
even the total loss of the fiber pig-tailed module is only 21%, which is an
appropriate level for generating squeezed vacuum. In contrast, the parametric
gain of this module is not very high ($\sim$15 dB when the pump power of 438
mW).

In contrast, OPA2 is fabricated by a chemical etching process, and its
thickness and width are about 5 $\mu$m and 5.6 $\mu$m. Because the waveguide
is relatively smaller than the one of OPA1, the loss is slightly higher, but a
parametric gain of 35 dB can be obtained for a pump power of 1.2 W. See the
references for more details (OPA1 Kashiwazaki2021 , OPA2 Kashiwazaki2020 ;
Kazama2021 ).

###  4.2 Effective efficiency

Equation 7 can be re-written as follows with efficiencies when $\theta=0$:

$\displaystyle\hat{I}_{\textrm{HD,OPA}}(t,0)\propto$
$\displaystyle\sqrt{\eta_{\textrm{HD}}\eta_{\textrm{OPA}}G}\hat{X}_{\textrm{HD}}(t)$
$\displaystyle+\sqrt{\eta_{\textrm{HD}}(1-\eta_{\textrm{OPA}})G}\hat{X}_{\textrm{vac,OPA}}(t)+\sqrt{1-\eta_{\textrm{HD}}}\hat{X}_{\textrm{vac,HD}}(t),$
(9)

where $\hat{X}_{\textrm{vac,OPA}}(t)$ and $\hat{X}_{\textrm{vac,HD}}(t)$ are
the quadrature-phase amplitude of the vacuum due to the losses. From this
equation, the effective efficiency $\eta_{\textrm{eff}}$ is calculated as

$\displaystyle\eta_{\textrm{eff}}=$
$\displaystyle\frac{\eta_{\textrm{HD}}\eta_{\textrm{OPA}}G}{\eta_{\textrm{HD}}\eta_{\textrm{OPA}}G+\eta_{\textrm{HD}}(1-\eta_{\textrm{OPA}})G+(1-\eta_{\textrm{HD}})}$
(10)
$\displaystyle=\frac{\eta_{\textrm{OPA}}\eta_{\textrm{HD}}}{\eta_{\textrm{HD}}+\frac{1-\eta_{\textrm{HD}}}{G}}$
(11) $\displaystyle\xrightarrow{G\to\infty}\eta_{\textrm{OPA}}.$ (12)

##  5 Data availability

Data are available from the authors upon request.

##  6 Acknowledgments

The authors acknowledge supports from the UTokyo Foundation and donations from
Nichia Corporation of Japan. T.Y. acknowledges support from the Advanced
Leading Graduate Course for Photon Science (ALPS). M.E. acknowledges support
from the Research Foundation for Opto-Science and Technology. This work was
partly supported by the Japan Science and Technology Agency (JPMJMS2064) and
the Japan Society for the Promotion of Science KAKENHI (18H05207, 20K15187).
The authors acknowledge Dr. Hiroshi Yamazaki for supporting the setup of the
broadband detection system, and Mr. Takeru Ebihara for fruitful discussion.

##  7 Author contributions

A.I. N.T., and T.U. conceived and planned the project. A.I., T. Kashiwazaki,
T.Y. and T.U. designed and constructed the experimental apparatus and acquired
the data. T. Kashiwazaki, T. Kazama, K.E., K.W., and T.U. developed the low-
noise and high-gain OPA module. A.I., T. Kashiwazaki, and M.E. analyzed the
data and wrote the manuscript with assistance from all other co-authors. A.F.
supervised the project.

##  8 Competing interests

The authors declare no competing financial interests.

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