11institutetext: Instituto de Astronomía Teórica y Experimental, CONICET-UNC,
Córdoba, Argentina.
11email<EMAIL_ADDRESS>22institutetext: Facultad de Matemática,
Astronomía, Física y Computación, Universidad Nacional de Córdoba (UNC),
Córdoba, Argentina.
22email<EMAIL_ADDRESS>33institutetext: Observatorio Astronómico de
Córdoba, UNC, Córdoba, Argentina.
33email<EMAIL_ADDRESS>44institutetext: Universidad Tecnológica
Nacional – Facultad Regional Mendoza, CONICET, CEDS, Rodriguez 243, Mendoza,
Argentina.

# Polarity relevance in flux-rope trajectory deflections
triggered by coronal holes

A. Sahade 112233 M. Cécere 1133 A. Costa 11 H. Cremades 44

(Received ; accepted )

###### Abstract

Context. Many observations suggest that coronal holes (CHs) are capable of
deviating the trajectory of coronal mass ejections (CMEs) away from them.
However, for some peculiar events, the eruption has been reported to be
initially pulled towards the CH and then away from it.

Aims. We study the interaction between flux-ropes (FRs) and CHs by means of
numerical simulations, with the ultimate goal of understanding how CHs can
deviate erupting CMEs/FRs from purely radial trajectories.

Methods. We perform 2.5D magnetohydrodynamical numerical simulations of FRs
and CHs interacting under different relative polarity configurations. In
addition, we reconstruct the 3D trajectory and magnetic environment of a
particular event seen by the STEREO spacecraft on 30 April 2012, whose
trajectory initially departed from the radial direction toward the CH but
later moved away from it.

Results. The numerical simulations indicate that at low coronal heights,
depending on the relative magnetic field polarity between FR and CH, the
initial deflection is attractive, i.e. the FR moves towards the CH (case of
anti-aligned polarities) or repulsive, i.e. the FR moves away from the CH
(case of aligned polarities). This is likely due to the formation of vanishing
magnetic field regions or null points, located between the FR and CH (case of
anti-aligned polarities) or at the other side of the FR (case of aligned
polarities). The analysed observational event shows a double-deflection
compatible with an anti-aligned configuration of magnetic polarities, which is
supported by SDO observations. We successfully reproduce the double deflection
of the observed event by means of a numerical simulation.

###### Key Words.:

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

##  1 Introduction

Large amounts of mass and magnetic field detachments are involved in the
release of coronal mass ejections (CMEs). Many CMEs show clear indications of
an embedded magnetic flux-rope structure (FR, i.e. magnetic field lines
twisted around an axial field). The FR magnetic system, usually associated
with filaments, frequently deflects from its outward radial direction. This
may happen due to multiple factors. For example, intrinsic CME and filament
properties may affect the final amount of deflection (e.g., Filippov et al.,
2001; Martin, 2003; Panasenco & Martin, 2008; Bemporad, 2009; Panasenco et
al., 2011; Pevtsov et al., 2012; Liewer et al., 2013; Möstl et al., 2015; Wang
et al., 2015; Kay et al., 2015, 2017). The magnetic field environment also
contributes to the deflection through the interaction of the CME/FR with
surrounding structures, for example: coronal holes (CH, e.g., Cremades et al.,
2006; Xie et al., 2009; Gopalswamy et al., 2009; Kilpua et al., 2009;
Panasenco et al., 2013), active regions (e.g., Kay et al., 2015; Möstl et al.,
2015), pseudostreamers (e.g., Lynch & Edmondson, 2013), streamer belts (e.g.,
Zuccarello et al., 2012; Kay et al., 2013; Yang et al., 2018), and
heliospheric current sheets (e.g., Liewer et al., 2015).

Previous reports have suggested that CME trajectories depend on the local and
global gradients of the magnetic pressure (Panasenco et al., 2013; Liewer et
al., 2015; Sieyra et al., 2020). They found that the analysed CMEs propagate
in the direction of least resistance, away from CHs. This behaviour was also
numerically studied by Sahade et al. (2020, hereafter S20), who found that the
FR deflection away from the CH was firstly due to the minimum magnetic energy
location and secondly, due to the channelling imposed by magnetic field lines.

Studies based on the evolution of CMEs in the high corona agree on that CHs
and open magnetic fluxes, acting as strong “magnetic walls”, repel
neighbouring CMEs by deflecting their trajectories (Cremades et al., 2006;
Gopalswamy et al., 2009; Gui et al., 2011; Yang et al., 2018; Cécere et al.,
2020). However, there are few peculiar examples for which CME paths behave
differently. Jiang et al. (2007) investigated a CME that first evolved toward
a CH and later moved away from it. Moreover, Sieyra et al. (2020, see their
figure 14) found that the trajectories of some of the analysed prominence-CMEs
were not always deflected away from CHs. Particularly at low coronal heights,
the eruption occasionally approached the CH, even when the trajectory was not
directed toward the region of minimum magnetic energy. Yang et al. (2018)
studied the interaction of a small filament with non-CH open field lines. The
filament polarity was anti-aligned with respect to the polarity of the open
magnetic field lines, meaning that the positive polarity footpoint of the
filament was adjacent to the negative polarity of the open magnetic fluxes.
They found that the erupted filament material first approached the open field
lines and afterwards was strongly deflected away from them.

To improve understanding of the interaction between CMEs and CHs we perform
2.5D MHD numerical simulations of a FR in the low corona. We analyse the
resulting trajectory deflection and we study the initial forces acting on the
FR (Section 2). We consider both, aligned and anti-aligned polarities of the
CH open field lines with respect to the nearest FR footpoint polarity. In
Section 3 we study an eruptive event that interacted with a CH, observed on 30
April 2012 by the Solar-Terrestrial Relations Observatory (STEREO, Kaiser et
al., 2008) twin spacecraft. It initially approached the CH and then propagated
away from it, similarly as the peculiar observed eruptive events mentioned
above. Conclusions and some open questions are presented in Section 4.

##  2 Numerical simulation

Figure 1: (a) Flux-rope and coronal hole scheme not to scale. The turquoise
frame indicates the simulated region. $h_{0}$ is the FR (orange circle) height
and $D$ is the FR distance to the CH (purple-shaded area) whose width is
parametrised by $W$. The line dipole and the image current are located at
depth $d$ and $h_{0}$, respectively. (b) Current wire layers: $r$ is the
radius, $\Delta$ is the thickness of the transition layer, $R_{-}$ is the
radial coordinate from the FR centre,$R_{+}$ is the radial coordinate from the
image wire , and $\phi$ the poloidal coordinate.

We perform numerical simulations to study the interaction between a FR and a
CH without other interfering magnetic structures. We use a simple 2.5
dimensional model that provides meaningful information about magnetic fields
and FR dynamics. The interaction is solved through the ideal MHD equations in
presence of a gravitational field. The equations in CGS units in the Cartesian
conservative form are written as:

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

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

Simulations were performed using the FLASH Code (Fryxell et al., 2000) in its
fourth version, operated with the USM (unsplit staggered mesh) solver, which
uses a second-order directionally unsplit scheme with a MUSCL-type
reconstruction. Outflow conditions (zero-gradient) are used at lateral and
upper boundaries, line-tied condition is used at lower boundary. To preserve
the initial force-free configuration outside the FR, a linear extrapolation of
the magnetic field is established (Zurbriggen, private communication). We can
neglect the magnetic resistivity and use the ideal MHD equations. This results
in a significant reduction of the computational cost, since the numerical
diffusion present in the simulations provides the necessary dissipation
(Krause et al., 2018). The highest resolution corresponds to $\sim[0.1\times
0.1]\leavevmode\nobreak\ \textrm{Mm}^{2}$ cells, in a
$[-700,700]\leavevmode\nobreak\ \textrm{Mm}\times[0,700]\leavevmode\nobreak\
\textrm{Mm}$ physical domain, where pressure and temperature gradients satisfy
the refinement criterion.

###  2.1 Magnetic model

We depart from an out-of-equilibrium magnetic FR, whose general configuration
is schematised in Fig. 1. The magnetic field of the FR is produced by a
current wire, an image current wire and a line dipole. The image current is
located below the photosphere having an opposite direction, so as to generate
a repulsive force. The line dipole, located below the photosphere, provides an
attractive force to the CME wire and emulates the photospheric field. The
equations are piecewise-defined in three zones:

Z1

Inside a current wire, $0\leq R<r-\frac{\Delta}{2}$.

Z2

Throughout the transition layer,
$r-\frac{\Delta}{2}\\!\leq\\!R\\!<r+\frac{\Delta}{2}$.

Z3

Outside a current wire, $r+\frac{\Delta}{2}\leq R$,

where $r$ is the current wire radius, $\Delta$ is the thickness of the
transition layer between the current wire and the exterior, and $R$ is the
radial coordinate from the centre of the current wire (Fig. 1b).

The magnetic field component $B_{\phi}$ generated by a current wire with
current distribution $j_{z}$ is given by:

$\\!\\!\\!\\!\\!\\!B_{\phi}(R)\\!=\\!\left\\{\begin{array}[]{rl}\begin{aligned}
&\tfrac{2\pi}{c}j_{0}R&&\text{\small at Z1}\\\ &\tfrac{2\pi
j_{0}}{cR}\left\\{\tfrac{1}{2}\left(r-\tfrac{\Delta}{2}\right)^{2}-\left(\tfrac{\Delta}{2}\right)^{2}+\right.\\\
&\tfrac{R^{2}}{2}+\tfrac{\Delta
R}{\pi}\text{sin}\left[\tfrac{\pi}{\Delta}\left(R-r+\tfrac{\Delta}{2}\right)\right]+\par\\\
&\left.\\!\\!\\!\left(\tfrac{\Delta}{\pi}\right)^{2}\cos\left[\tfrac{\pi}{\Delta}\left(R-r+\tfrac{\Delta}{2}\right)\right]\right\\}\qquad&&\text{\small
at Z2}\\\ &\tfrac{2\pi
j_{0}}{cR}\left[r^{2}+\left(\tfrac{\Delta}{2}\right)^{2}-2\left(\tfrac{\Delta}{\pi}\right)^{2}\right]&&\text{\small
at Z3,}\end{aligned}\end{array}\right.$ (6)
$\\!j_{\mathrm{z}}(R)\\!=\\!\left\\{\begin{array}[]{rl}\begin{aligned}
&\\!j_{0}&&\text{\small at Z1}\\\
&\\!\tfrac{j_{0}}{2}\left\\{\cos\left[\tfrac{\pi}{\Delta}\left(R-r+\tfrac{\Delta}{2}\right)\right]+1\right\\}\quad&&\text{\small
at Z2}\\\ &\\!0&&\text{\small at Z3;}\end{aligned}\end{array}\right.$ (7)

where $j_{0}$ is a current density.

In order to obtain a helical magnetic field in the FR we include a magnetic
field in the $z$-axis of strength $B_{\mathrm{z}}$. In this way, as we showed
in S20, we avoid excessive gas pressure values needed to balance the magnetic
pressure inside the flux-rope in the initial equilibrium state. The component
$B_{\mathrm{z}}$ of the magnetic field and the current distribution
$j_{\phi}$, are described by:

$B_{\mathrm{z}}(R)=\tfrac{\sqrt{8}\pi
j_{1}}{c}\sqrt{\left(r-\tfrac{\Delta}{2}\right)^{2}-R^{2}}\,,$ (8)
$j_{\phi}(R)=j_{1}R\left[\sqrt{\left(r-\tfrac{\Delta}{2}\right)^{2}-R^{2}}\right]^{-1}\,,$
(9)

where $j_{1}$ is a current density. These expressions are valid inside the
flux-rope (Z1) and are null in the rest of the domain.

For the initial magnetic field of the CH we use the same expressions as in
S20. The total initial magnetic field is the sum of the magnetic field of the
FR and the magnetic field of the CH, which in Cartesian components is given
by:

$\displaystyle B_{x}=$ $\displaystyle B_{0}\
\sin\left(\frac{x-D}{W}\right)\,\exp[-y/W]-B_{\phi}(R_{-})\tfrac{(y-h_{0})}{R_{-}}+$
$\displaystyle\>B_{\phi}(R_{+})\tfrac{(y+h_{0})}{R_{+}}-MdB_{\phi}{\scriptstyle\left(r+\tfrac{\Delta}{2}\right)}\left(r+\tfrac{\Delta}{2}\right)\tfrac{x^{2}-(y+d)^{2}}{R_{d}^{4}}\,,$
$\displaystyle B_{y}=$ $\displaystyle B_{0}\
\cos\left(\frac{x-D}{W}\right)\,\exp[-y/W]+B_{\phi}(R_{-})\tfrac{x}{R_{-}}-$
$\displaystyle
B_{\phi}(R_{+})\tfrac{x}{R_{+}}-MdB_{\phi}{\scriptstyle\left(r+\tfrac{\Delta}{2}\right)}\left(r+\tfrac{\Delta}{2}\right)\tfrac{2x(y+d)}{R_{d}^{4}}\,,$
$\displaystyle B_{z}=$ $\displaystyle B_{\text{z}}(R_{-})\,.$ (10)

where the parameter $B_{0}$ is the radial magnetic field strength of the CH at
the distance $D$ on the $x$-axis. The parameter $W$ is related to the width of
the CH (see next subsection) and modifies the decay of the magnetic field
strength in the $y$-direction. $h_{0}$ is the initial vertical position of the
FR and $M$ is the intensity of the line dipole at depth $d$. The distances $R$
are:

$R_{\pm}=\sqrt{x^{2}+(y\pm h_{0})^{2}},$ $R_{d}=\sqrt{x^{2}+(y+d)^{2}},$

where $R_{-},R_{+}$ and $R_{d}$ originate in the FR centre, image current wire
and line dipole, respectively (see Fig. 1 of S20 for more details).

###  2.2 Thermodynamic variables

We simulate the solar atmosphere by adopting a multi-layer structure (Mei et
al., 2012). The chromosphere lies between $y=0$ and $y=h_{\mathrm{ch}}$ with
constant temperature $T_{\mathrm{ch}}$. The transition region, located between
$y=h_{\mathrm{ch}}$ and the base of the corona ($y=h_{\mathrm{c}}$), is
represented by a linearly increasing temperature up to $T_{\mathrm{c}}$, which
is the constant temperature assumed for the corona. Thereby, the initial
temperature distribution is given by

${\textstyle T(y)=}\left\\{\begin{array}[]{rl}\begin{aligned} &{\textstyle
T_{\mathrm{ch}}}&&\text{\small if $0\leq y<h_{\mathrm{ch}}$}\\\
&{\textstyle(T_{\mathrm{c}}-T_{\mathrm{ch}})\left[\frac{y-h_{\mathrm{ch}}}{h_{\mathrm{c}}-h_{\mathrm{ch}}}\right]+T_{\mathrm{ch}}}\quad&&\text{\small
if $h_{\mathrm{ch}}\leq y<h_{\mathrm{c}}$}\\\ &{\textstyle
T_{\mathrm{c}}}&&\text{\small if $h_{\mathrm{c}}\leq y$
}.\end{aligned}\end{array}\right.$ (11)

We set a temperature of $T_{\mathrm{ch}}=10\,000\,\text{K}$ for the
chromosphere and $T_{\mathrm{c}}=10^{6}\,\text{K}$ for the corona. The height
of the chromosphere is $h_{\mathrm{ch}}=10\,\text{Mm}$, while the base of the
corona is at $h_{\mathrm{c}}=15\,\text{Mm}$.

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

$\\!T(R_{-})\\!=\\!\left\\{\begin{array}[]{rl}\begin{aligned}
&\\!T_{\text{\tiny{FR}}}&&\text{\small at Z1}\\\
&\\!\\!(T_{\mathrm{c}}\\!-\\!T_{\text{\tiny{FR}}})\\!\left[\tfrac{R_{-}-(r+\Delta/2)}{\Delta}\right]\\!+\\!T_{\text{\tiny{FR}}}\quad&&\text{\small
at Z2}\\\ &\\!T_{\mathrm{c}}&&\text{\small at
Z3.}\end{aligned}\end{array}\right.$ (12)

In the previous work (S20), we considered the current-free atmosphere in
hydrostatic equilibrium. Hence, the pressure $p(y)$ was only a function of $y$
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 Sun mass, $R_{\sun}$ is the
solar radius, and $y=0$ corresponds to the solar surface). In this work we add
a dimensionless factor $\chi$ to the pressure distribution, to set a subdense
CH (it is based on the density distributions proposed by Pascoe et al., 2014)

$\\!\chi(x,y)\\!=\\!\left\\{\begin{array}[]{rl}\begin{aligned}
&\\!\beta(x,y)+1&&\text{\small if $y<W/2$}\\\
&\\!\left(2-2\frac{y}{W}\right)\,\beta(x,y)+1\quad&&\text{\small if $W/2\leq
y<W$}\\\ &\\!1&&\text{\small if $W\leq y$,}\end{aligned}\end{array}\right.$
(13)

where

$\beta(x,y)=\left(\frac{n_{\mathrm{CH}}}{n_{\mathrm{c}}}-1\right)\mathrm{sech}^{2}\left(\frac{x-D}{W\arcsin{(0.25\exp[y/W}])}\right)^{2}\,.$

The number density at height $y=h_{\mathrm{c}}$ in the corona is
$n_{\mathrm{c}}$, while inside the CH the number density decays to
$n_{\mathrm{CH}}$. According to the resulting subdense zone, the effective
width of the CH is set to $w\sim W/2$. Then the pressure is
$P(x,y)=\chi(x,y)p(y)$, where $p(y)$ is the hydrostatic component:

${\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]}}&&\text{\small
if $0\leq y<h_{\mathrm{ch}}$}\\\
&{\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&&\text{\small
if $h_{\mathrm{ch}}\leq y<h_{\mathrm{c}}$}\\\
&{\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]}}&&\text{\small
if $h_{\mathrm{c}}\leq y$,}\par\end{aligned}\end{array}\right.$ (14)

where

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

$\alpha=\frac{m_{i}GM_{\sun}}{2k_{B}}$, and $N_{A}$ is the Avogadro number.
The internal pressure of the FR is obtained by proposing a solution close to
the equilibrium:

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

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

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

###  2.3 Setup

To perform a 2.5D simulation we assume that the FR and the CH have a symmetry
in the $z$-direction. A FR of characteristic length $L_{0}\sim
500\,\textrm{Mm}$ (Berger, 2014) is large enough to make the symmetry
assumption appropriate. Given this characteristic length $L_{0}$, the CHs
result in areas ($A\sim wL_{0}$) between $7.5\times 10^{4}\,\text{Mm}^{2}$ and
$1.5\times 10^{5}\,\text{Mm}^{2}$ (Hofmeister et al., 2017; Heinemann et al.,
2019).

We select the same model parameters as in S20 (see their Sec. 2.5 for more
details). We vary the CH parameters as in Table 1 to obtain a set of 18
simulations, which are run with two different FRs to obtain more general
results. The initial parameters describing FR1 and FR2 simulations are listed
in Table 2.

Table 1: CH parameters.

Case | $B_{0}\,$ | $D\,$ | $W\,$
---|---|---|---
| (G) | (Mm) | (Mm)
1 | $0.4$ | $150$ | $400$
2 | $-0.4$ | $150$ | $400$
3 | $0.8$ | $150$ | $400$
4 | $-0.8$ | $150$ | $400$
5 | $1.2$ | $150$ | $400$
6 | $-1.2$ | $150$ | $400$
7 | $1.6$ | $150$ | $400$
8 | $-1.6$ | $150$ | $400$
9 | $0.8$ | $250$ | $400$
10 | $-0.8$ | $250$ | $400$
11 | $0.8$ | $350$ | $400$
12 | $-0.8$ | $350$ | $400$
13 | $0.8$ | $150$ | $300$
14 | $-0.8$ | $150$ | $300$
15 | $0.8$ | $150$ | $500$
16 | $-0.8$ | $150$ | $500$
17 | $0.8$ | $150$ | $600$
18 | $-0.8$ | $150$ | $600$
111Parameter $B_{0}$ is the CH magnetic field strength, $D$ its distance to
the FR, and $W$ is related to its width.

Cases in Table 1 with positive (negative) magnetic strength $B_{0}$ are
aligned (anti-aligned), meaning that the FR footpoint which is nearest to the
CH has positive (positive) polarity, as shown in the left (right) top panel of
Fig. 2. Note that the left panels of this figure are equivalent configurations
(aligned), as are those on the right (anti-aligned). It is the relative
polarity alignment what provides the different scenarios. The cases in the two
upper panels represent the two possible relative FR-CH polarities.

Table 2: Initial state parameters.

Parameter | Value
---|---
| FR1 | FR2
$j_{0}\,(\text{stA}\,\text{cm}^{-2})$ | $435$ |  $295$aa$a$ The value $j_{0}$ for FR2 was corrected because of a typo error in S20.
$j_{1}\,(\text{stA}\,\text{cm}^{-2})$ | $455$ | $300$
$T_{\text{\tiny{FR}}}\,(\text{MK})$ | $1$ | $4$
$n_{\mathrm{c}}\,(\text{cm}^{-3})$ | $3\times 10^{8}$ | $4.5\times 10^{8}$
$n_{\mathrm{CH}}\,(\text{cm}^{-3})$ | $2\times 10^{8}$
$h_{0}\,(\text{Mm})$ | $30$
$r\,(\text{Mm})$ | $2.5$
$\Delta\,(\text{Mm})$ | $0.25$
$d\,(\text{Mm})$ | $3.125$
$M$ | $1$
222Parameters $j_{0}$ and $j_{1}$ are the current densities inside the flux-
rope in the $z$-direction and $\phi$-direction, respectively,
$T_{\text{\tiny{FR}}}$ is the internal FR temperature, $n_{\mathrm{c}}$ is the
numerical density at the base of the corona, $n_{\mathrm{CH}}$ is the
numerical density for the CH, $h_{0}$ is the vertical position (height) of the
FR, $r$ is its radius, and $\Delta$ is the thickness of the transition layer
between the FR interior and the corona. Parameters $d$ and $M$ are the depth
of the line dipole below the boundary surface and its relative intensity,
respectively.

Figure 2:  Relative polarity scheme. The left panels show cases with aligned
FR and CH polarities, for which a null point (green circle) forms on the
opposite of the CH. Right panels show the anti-aligned configuration, for
which the minimum forms between the FR and CH. The forces $F$ acting on the FR
centre are directed towards the null point. The green $\times$ symbols
indicate the site of reconnection.

###  2.4 Results

Figure 3: Paths of the FR centre as a function of time (dashed line: aligned
case, dotted line: anti-aligned case). The purple star indicates the initial
position of the FR centre, the blue circles the initial position of the
magnetic null point ($E_{\textrm{min}}$) and the blue arrows the magnetic
force direction on the FR ($F_{\textrm{m}}$) at $t=0\,$s.

As we mentioned in S20, a minimum magnetic energy region (hereafter null
point) arises at the position where the magnetic field of the CH and FR
counteracts (green circles of Fig. 2). The FR-CH aligned polarity cases with
positive $B_{0}$, have the null point to the left of the FR (negative
$x$-positions, see top left panel of Figure 2); while the FR-CH anti-aligned
polarity cases with negative $B_{0}$, have the null point to the right of the
FR (positive $x$-positions, see top right panel of Figure 2). We note that the
anti-aligned cases have the null point between the FR and the CH. For both
configurations, we calculate the forces acting on the FR centre at $t=0\,$s.
We find that the magnetic force is the main contribution to the total force.
The magnetic force of the simulated cases always points towards the magnetic
null point, initially deflecting the FR towards this position. In the aligned
cases the magnetic force points towards a direction away from the CH. The
opposite occurs for the anti-aligned cases.

Figure 3 displays for FR2 the aligned case 7 (negative $x$ values) and the
anti-aligned case 8 (positive $x$ values). In the figure we show the initial
null point locations (blue circles), the initial magnetic forces on the FR
centre (cyan arrows) and the resulting trajectories. The last point plotted
corresponds to the time when the trajectory becomes parallel to the magnetic
field: $700\leavevmode\nobreak\ \textrm{s}$ for case 7 and
$1200\leavevmode\nobreak\ \textrm{s}$ for case 8. The trajectory of case 7
(dashed-line) deflects only once to the left, while that of case 8 (dotted-
line) is first deflected to the right and later to the left, i.e. the FR
suffers a double deflection. The general dynamics of the aligned cases are
represented by case 7, while the anti-aligned cases are represented by case 8.
It follows that the single or double deflection only depends on the relative
alignment of the FR and CH polarities.

Figure 4: Initial magnetic energy distribution and magnetic field lines for
case 8. The dotted line represents the path of the FR centre as a function of
time until $t=1200\leavevmode\nobreak\ \textrm{s}$. The animated evolution of
this figure is available in the HTML version.

To explain the double-deflection mechanism seen in anti-aligned cases, we
analyse the evolution of the magnetic energy density, also considering the
magnetic field lines. Figure 4 (see animated version available in the HTML
version) displays the initial magnetic energy distribution and the magnetic
field lines for case 8. The maximum energy (red in the colour scale) is
located at the FR centre and the minimum (blue in the colour-scale) is located
between the FR centre and the CH. The background energy decays with height,
along the $y$-axis. Initially the FR travels towards the null point and
collides with it. Then the null point stretches, leading to a low magnetic
energy region that surrounds the FR front. A turbulent zone is excited in this
low magnetic energy region, whose plasma parameter $\beta\approx 1$,
entangling the magnetic field lines. The FR continues moving in the original
direction, driving a coronal shock wave that compresses and bends the field
lines. This results in a magnetic force that eventually slows down the lateral
motion of the FR, redirecting its trajectory to finally become parallel to the
field lines. In summary, in the low corona and for an anti-aligned case, the
magnetic force appears as the main driver of the FR trajectory evolution: it
is firstly attracted toward the null point, then it surpasses the null point
position due to inertia, until the magnetic force exerted by the compressed
and bent CH magnetic field lines pushes the FR away. The final FR path is that
of least resistance, i.e. it travels guided by the magnetic field lines
towards a region of less magnetic energy. Thus, at later stages of evolution
all FRs move away from the CH, independently of the polarity configuration,
guided by the magnetic field.

Figure 5: a) Null point locations at $t=0\,$s for all cases. The purple star
represents the position of the FR, black edges correspond to FR1 simulations
and grey edges to FR2 ones. Dashed lines indicate the location of null points
for aligned (left) and anti-aligned (right) cases, with the CH at the same
distance $D$ and same parameter $W$. The symbol colour changes with the
strength $B_{0}$ of the magnetic field (from weaker to stronger: pink, light
blue, blue violet and dark blue), the size changes with the parameter $W$
(bigger means wider), and the shape changes with the distance $D$ (from closer
to farther: circle, triangle and square). b) Module of the magnetic force per
unit length versus null point distance to the FR centre. Symbols hold the same
properties as in a). A black dot in the symbol centre indicates an anti-
aligned case.

Figure 5a shows the null point locations for all cases in Table 1. The purple
star represents the location of the FR centre. The symbols with black and grey
edges correspond to FR1 and FR2 simulation cases, respectively. The colour of
the symbols changes with the strength of the magnetic field $B_{0}$: dark blue
for stronger ($1.6\leavevmode\nobreak\ \textrm{G}$) and pink for weaker
($0.4\leavevmode\nobreak\ \textrm{G}$). The shape of the symbols change with
the distance between the CH and the FR: circle ($D=150\leavevmode\nobreak\
\textrm{Mm}$), triangle ($D=250\leavevmode\nobreak\ \textrm{Mm}$) and square
($D=350\leavevmode\nobreak\ \textrm{Mm}$). Changes in the CH parameter $W$ are
represented with different symbol sizes: the smallest for
$W=300\leavevmode\nobreak\ \textrm{Mm}$ and the largest for
$W=600\leavevmode\nobreak\ \textrm{Mm}$.

We note that for a given parameter $W$, at a given distance $D$, there is a
linear correlation between the location of the null point and the magnetic
field strength $B_{0}$. The larger the magnetic field strength, the nearest
the null point is to the FR centre. In Fig. 5a the CH cases with
$W=400\leavevmode\nobreak\ \textrm{Mm}$ (same size symbols) and distance
$D=150\leavevmode\nobreak\ \textrm{Mm}$ (circle symbols) generate the dashed
grey lines of null point positions. The straight line with positive (negative)
slope represents the cases with anti-aligned (aligned) polarity.

Also, for a given distance $D$ (same symbol shape) and a given magnetic field
strength $B_{0}$ (same colour), a change in $W$ (different sized circles)
yields a curve that crosses the straight lines. For the aligned polarity cases
the larger the CH width, the higher the null point position. Conversely, for
the anti-aligned cases the larger the CH width, the lower the height of the
null point.

Moreover, for a given magnetic field strength $B_{0}$ (same symbol colour) and
a given parameter $W$ (same symbol size) a change in the distance (different
symbols) yields a curve that crosses the straight line. For the aligned
polarity cases the larger the distance, the lower the height of the null
point. On the contrary, for the anti-aligned cases the larger the distance,
the higher the null point location.

In Fig. 5b a positive correlation between the magnetic force magnitude per
unit length and the null point distance (d) to the FR can be noticed. This
correlation relies on the magnetic strength $B_{0}$, resulting in four
different slopes (linking both polarities and FR configurations); the larger
the magnetic field strength, the larger the slope. The anti-aligned cases are
differentiated from the aligned ones by a black dot in the symbol centre.

This analysis sheds light on the relationships between null point location,
distance between CH and FR, strength of the CH magnetic field and CH area. A
deeper study that considers a wider range of parameters is required, so as to
predict null point locations more precisely.

##  3 An observational case study

Figure 6: Tie-pointing reconstruction of the 30 April 2012 prominence. The
three white plus symbols in the top panels mark the wide apex considered in
STEREO EUV images. The red plus symbol in the bottom panels show the apex in
the STEREO coronagraph images.  Figure 7: 3D evolution of the prominence apex
since 04:06 UT to 13:30 UT, from bluish to reddish spheres. The grey straight
line represents the radial direction at the initial position of the apex. Blue
lines belong to the open field lines of the CH and the white ones are the
closed magnetic field lines. Red meridian points at the Earth direction, green
meridian at the STEREO B direction, and the light-blue line marks the solar
Equator.

###  3.1 Methods and techniques

To compare our model with observations, we analyse the trajectory of an
eruptive prominence associated with a CME that occurred on April 30, 2012.
During the outward motion of the prominence, the trajectory changed its
direction twice, i.e. the eruption suffered a double deflection. A nearby CH,
together with the eruptive prominence, constitute a relatively isolated
scenario that can be approximated by one of the schemes in Fig. 2. The
eruption occurred to the east of the northern hemisphere starting at 04:06 UT.
To reconstruct the 3D trajectory of the eruption we track the prominence cold
material by means of EUV images from the Extreme-Ultraviolet Imager (EUVI) of
the SECCHI instrument suite (Sun-Earth Connection Coronal and Heliospheric
Investigation, Howard et al. 2008) and white-light images from the SECCHI COR1
coronagraphs on board the STEREO A (ST-A) and STEREO B (ST-B) spacecraft. From
04:06 UT to 11:56 UT we use EUVI $304\,$Å images from ST-A and ST-B. Running
difference images from COR1 on board ST-A and ST-B are used to track the
eruption at a later stage, from 11:45 UT to 13:30 UT.

The position of the prominence in time is determined by means of the tie-
pointing reconstruction technique (hereafter triangulation, Inhester, 2006;
Mierla et al., 2010), which is applied to the apex, chosen to represent the
global motion of the ejection. To allow for a better description of the
trajectory, we consider a wide apex by triangulating three points from the
prominence front at each analyzed time (see plus symbols in Figure 6, top
panels). The sequences of the three points follow the same trend, therefore,
we consider that they represent well the global motion of the ejection. We
tracked other points of the prominence and we found that they also follow the
same trend. In addition, we triangulate the prominence feet positions to
estimate their location and length. Figure 6 upper panels show, in addition to
the points characterising the apex (white plus symbols), those representing
the prominence feet (pink dot symbols) in the $304\,$Å filter of EUVI-A and
EUVI-B; while the lower panels of the figure show the prominence apex in
COR1-A and COR1-B.

To analyse the magnetic structure surrounding the ejection area we use the
Potential Field Source Surface (PFSS) model by Schrijver & De Rosa (2003).

###  3.2 Prominence evolution: trajectory and magnetic environment

The 3D trajectory determined by applying the triangulation method is displayed
in Fig. 7. As can be seen from the figure, the eruption initially approaches
the negative open magnetic field lines (blue lines, reconstructed from PFSS)
and then moves away from it, following a new direction. The open magnetic
field lines correspond to a CH that was catalogued as SPoCA 4548 (Barra et
al., 2009; Verbeeck et al., 2014) when it was facing Earth on 4 April 2012. We
measure the northeast and southwest feet of the erupting prominence on 30
April 2012 at $t=05:06\leavevmode\nobreak\ \textrm{UT}$ and at a height
$R=1.09\,R_{\odot}$, to be located at $(67^{\circ},264^{\circ})$ and
$(60^{\circ},307^{\circ})$ respectively. The initial measured height, average
latitude and Carrington longitude of the prominence are
$(R_{0},\theta_{0},\phi_{0})=(1.12\,R_{\odot},61.6^{\circ},289.4^{\circ})\,$
at 04:06 UT (cyan point in Fig. 7 and cyan star in Fig. 8). Then, the
prominence apex moves towards the closest minimum magnetic energy region,
approaching the CH until
$(R_{cl},\theta_{cl},\phi_{cl})=(1.45\,R_{\odot},59.4^{\circ},278.1^{\circ})\,$
at 10:46 UT (orange points in Fig. 7). Next, the prominence apex deflects away
from the CH towards the heliospheric current sheet and the final measured
position is
$(R_{f},\theta_{f},\phi_{f})=(2.67\,R_{\odot},61.4^{\circ},336.8^{\circ})\,$
at 13:30 UT (higher red point in Fig. 7 and red star in Fig. 8). Thus, the
prominence suffers a double deflection: first it moves towards lower
longitudes when approaching the CH (from $289^{\circ}$ to $278^{\circ}$) and
afterwards it moves away from it, increasing its longitude from $278^{\circ}$
to $336^{\circ}$ and reaching the heliospheric current sheet.

The prominence erupts under the scenario described in Figure 8. The figure
shows a map of the magnetic field strength between 200∘ and 340∘, where we
indicate the magnetic structures as catalogued by the National Oceanic and
Atmospheric Administration (NOAA) and Wilcox Solar Observatory (WSO) Source
Surface Synoptic Charts. The pink shaded areas enclose the CHs, with size and
location estimated from the open lines of the PFSS and the dark regions of the
ST-B/EUVI $195\,$Å filter. The cyan star indicates the source region (SR), the
orange star indicates the closest position to the CH, and the red star shows
the last position measured. Several active regions can be noticed in the map.
Note that, as a result of the projection, AR 11467 may appear to be at a
similar distance from the SR as the CH to the northeast. But in fact, the
nearest region of that AR (namely its eastern extension in the cyan colour) is
almost two times farther away than the nearest edge of the CH. The dashed line
represents the heliospheric current sheet. Between the CH and the SR there is
a zone of low magnetic strength (between the cyan and orange star). Figure 8
also shows a region of even lower magnetic strength towards the north and the
southwest of the SR.

As the eruption took place at very high latitudes ($\sim\,60^{\circ}$) and
close to the limb from Earth’s perspective, the characterisation of the
photospheric magnetic field was not straightforward. The identification of the
polarities at both sides of the neutral line was based on the following
reasoning, arising from Figure 9. The figure depicts SDO/AIA coronal
wavelengths overlapped to an SDO/HMI magnetogram, both at a time before
eruption of the prominence. The prominence, surrounded by a circular coronal
cavity, is seen above the north-northeast limb. At a later time, well after
the eruption (starting ca. 12:00 UT), post eruptive loops can be discerned
particularly in SDO/AIA 211 Å. These loops are not seen at the time of Figure
9, but the approximate region where they form is encircled in the figure. In
fact, the loops form above the polarity inversion line seen as a dark channel
ending at the prominence location, close to the limb. Therefore, it appears
highly plausible that the eruptive prominence was sitting at that same neutral
line, which enables identification of the magnetic field polarity at its
sides. According to the information provided by the SDO/HMI magnetogram, there
is predominance of negative polarity to the northwest of the neutral line
holding the prominence, and positive polarity to the southeast. This scenario,
together with the fact that the magnetic field polarity of the CH is negative,
suggests that this event is an example of an anti-aligned case.

Figure 8: Magnetic strength map at $R=1.12\,R_{\odot}$ (initial height of the
prominence), from longitude $200^{\circ}$ to $340^{\circ}$, on 30 April 2012
at 00:00 UT. The stars represent the initial (cyan), the closest to CH
(orange), and the last measured (red) prominence position. The CHs, the active
regions (NOAA) and the heliospheric current sheet (WSO) are drawn over the
map. PCH stands for polar coronal hole. The map has been provided by the
Community Coordinated Modeling Center at Goddard Space Flight Center through
their public Runs on Request system (http://ccmc.gsfc.nasa.gov) with the PFSS
model.  Figure 9: SD0/AIA composite image of $211\,$Å, $193\,$Å and $171\,$Å
at 01:26 UT overlapped to an SDO/HMI image at 01:30 UT on the day of the
event. In the latter, magenta corresponds to magnetic fields with positive
polarity and blue to negative ones. The prominence and its overlying cavity,
as well as the polarity inversion line (PIL) were the prominence presumably
lies are indicated by arrows. The white circle denotes the region were post-
eruptive loops form at a later time after eruption.

###  3.3 Simulated event

Figure 10: Height versus average longitudinal deflection of the 30 April 2012
prominence (dots) and the trajectory of the simulated FR (solid line) with the
purple star showing the initial FR centre position. Table 3: Parameters used
for the simulation of the observed event.

Parameter | Values
---|---
$j_{0}\,(\text{stA}\,\text{cm}^{-2})$ | $435$
$j_{1}\,(\text{stA}\,\text{cm}^{-2})$ | $455$
$T_{\text{\tiny{FR}}}\,(\text{MK})$ | $0.8$
$n_{\mathrm{c}}\,(\text{cm}^{-3})$ | $3\times 10^{8}$
$n_{\mathrm{CH}}\,(\text{cm}^{-3})$ | $2\times 10^{8}$
$h_{0}\,(\text{Mm})$ | $40$
$r\,(\text{Mm})$ | $2.5$
$\Delta\,(\text{Mm})$ | $0.25$
$d\,(\text{Mm})$ | $3.125$
$M$ | $1$
$B_{0}\,(\text{G})$ | $-0.6$
$D\,(\text{Mm})$ | $300$
$W\,(\text{Mm})$ | $400$
333Parameters $j_{0}$ and $j_{1}$ are the current densities inside the flux-
rope in $z$-direction and in $\phi$-direction, respectively,
$T_{\text{\tiny{FR}}}$ is the internal FR temperature, $n_{\mathrm{c}}$ the
numerical density at the base of the corona, $n_{\mathrm{CH}}$ the numerical
density for the CH, $h_{0}$ the vertical location (height) of the FR, $r$ its
radius, and $\Delta$ the thickness of the transition layer between the FR
interior and the corona. Parameters $d$ and $M$ are the depth of the line
dipole below the boundary surface and its relative intensity, respectively.
The CH parameters are $B_{0}\,$, $D\,$ and $W$.

Figure 11: Synthetic EUV emission for filters $304\,$Å and $171\,$Å using the
AIA response temperature. White dotted-lines represent the estimated boundary
of the CH. The animation of the full evolution is available at the HTML
version.

From the observational event analysed in the previous section, a question
arises: is it possible that the mere presence of the CH results in the
attraction of the FR structure? The numerical study carried out here together
with information on the magnetic field configuration suggests that this may be
the case if the FR and CH are magnetically anti-aligned.

Considering that the latitudinal displacement is negligible in comparison to
the longitudinal one, we perform a simulation to emulate the deflection in the
latter direction. The axis of the prominence is oriented along the
northeast–southwest direction and its length is $\sim 300\leavevmode\nobreak\
\textrm{Mm}$, which is large enough to assume a 2.5D approximation. Our model
simulates the transverse motion projected on the longitudinal coordinate.
Table 3 lists the parameters used for the simulation. We set the temperature
$T_{\mathrm{FR}}$ to emulate the emission of the prominence in the
corresponding EUV filters. The CH parameters are estimated from the magnetic
maps ($B_{0}$) and from the EUVI-B $195\,$Å filter ($D$ and $W$).

Figure 10 shows the average longitudinal deflection for the triangulated event
(coloured dots), and the deflection of the simulated case (solid line). We
note that a model of anti-aligned polarity is able to reproduce the deflection
of the observed event. Taking into account that many effects are not
considered in this simple model, the match between the simulation and the
observation suggests that the proposed magnetic configuration is a possible
cause of the prominence approach to the CH.

Figure 11 shows the synthetic emission in the $304\,$Å and $171\,$Å
wavelengths generated by the FoMo tool (Van Doorsselaere et al., 2016) at
$t=200\,$s. The code sets the thickness of simulation to 1 Mm since it is
2.5D. The subdense CH is set to the left of the FR structure, while the
simulated FR is seen to emit in both wavelengths. The full evolution can be
seen as an animation in the HTML version. During its evolution a subdense
cavity is formed around the FR (clearly noticed in the 171 Å filter). The
movie also shows the formation of waves and shocks.

##  4 Conclusions

Determining the mechanisms and the scenarios that produce deflections during
the evolution of CMEs is a crucial step for improving space weather
predictions. Many studies suggest that open magnetic fluxes behave as
“magnetic walls” that push CMEs away (e.g., Cremades et al., 2006; Gopalswamy
et al., 2009; Gui et al., 2011; Cécere et al., 2020). However, there are few
reported events where the eruptions first approach open magnetic fluxes and
then propagate away from them (Jiang et al., 2007; Yang et al., 2018; Sieyra
et al., 2020). In these observational cases the CME structure is close to, not
only a CH, but also other magnetic structures, such as an active region or a
pseudostreamer, making it difficult to distinguish which of all the structures
are responsible for the double deflection and to what extent. Therefore, it is
important to study cases in which the FR-CH system is almost isolated, both
observationally and numerically.

In our previous work (S20) we studied the FR-CH interaction numerically,
simulating cases with aligned polarities between the two structures, and
showing the importance of the null magnetic region to predict the FR early
evolution. Here, we used the same 2.5D model for which the FR-CH interaction
could be studied in isolation. Through numerical simulations we found that the
FRs evolve towards the null magnetic region whose position is determined by
the FR and CH parameters. Particularly, the site where the null region is
formed depends on the relative polarity alignment between FR and CH. During
the early evolution of the system, the null point attracts the FR, in the
aligned cases deflecting it away from the CH and in the anti-aligned cases
causing that the FR approaches the CH. However, as in many observational cases
mentioned above, in their later evolution all FRs move away from the CH,
guided by the magnetic field, independently of the relative FR-CH polarity
alignment. The physical mechanism is the same for both alignment cases, but
the phenomenological result is different: in the low corona, the anti-aligned
cases will exhibit a double-deflection trajectory, while the aligned cases
will display a single deflection trajectory.

Performing several numerical simulations of different cases, varying FR
properties and CH parameters, we found:

  1. 1.

For a given CH width, at a given distance to the FR, there is a linear
correlation between the position of the null point and the magnetic field
strength. The larger the magnetic field strength, the nearest the null point
to the FR centre.

  2. 2.

In the aligned (anti-aligned) cases, for a given distance and a given magnetic
field strength, the larger the width the higher (lower) the null point
location.

  3. 3.

In the aligned (anti-aligned) cases, for a given magnetic field strength and a
given width, the larger the distance the lower (higher) the null point
location.

  4. 4.

There is a positive correlation between the magnetic force magnitude per unit
length and the null point distance to the FR. This correlation depends on the
magnetic field strength: the larger the magnetic field strength, the larger
the slope of the correlation.

To compare our model with observations we analyse an event occurred on 30
April 2012. We consider the system as a relatively magnetically isolated
scenario, where an ejected prominence is attracted by a nearby CH before it is
deflected away from it. A careful analysis of AIA low coronal wavelengths,
together with HMI photospheric magnetic fields (see Fig. 9) reveals the
magnetic polarities at the sides of the eruptive prominence. The analysis
indicates that the magnetic polarities of the prominence source and of the CH
are configured in an anti-aligned fashion. Assuming this particular alignment
between both structures in a simple numerical model (where many effects were
not considered) and approximate parameters estimated from the observations, we
were able to reproduce the measured longitudinal deflection of the analysed
event. This supports the conjecture that the presence of the CH and the
magnetic topology that it produces are responsible for the double deflection
of the event. We can thus suggest that the null point first attracts the
prominence towards the CH until the structure of the open magnetic fields
deflects it away. Finally, the prominence follows the least resistance path
towards the heliospheric current sheet.

Although, as shown in the literature, the scenarios in which the polarities
are anti-aligned (see Sec. 2.3) seem to be common, most studies show that CMEs
are deflected away from CHs. However, these facts are not necessarily
incompatible. On one hand, most of the observational events analysed in the
literature are restricted to the high coronal region, so that a CME could have
approached a CH at an early stage and thus, the double deflection would have
not been reported. On the other hand, in studies that include the low corona,
the deflection toward a CH may have been interpreted as the action of other
magnetic structures pulling the CME towards the CH, without considering the
mere presence of the CH as the possible cause of the attraction. Some
questions arise from this new insight: How common is the anti-aligned
configuration? Is there a relation between the deflection rate and the FR-CH
alignment? To answer these questions, more FR-CH interactions should be
studied in detail at low coronal heights.

###### Acknowledgements.

We thank the reviewer, Dr. Paulett Liewer, for her valuable suggestions that
helped us to substantially improve the previous version of this manuscript. AS
is doctoral fellow of CONICET. MC, AC and HC are members of the Carrera del
Investigador Científico (CONICET). AS, MC and AC acknowledge support from
ANPCyT under grant number PICT No. 2016-2480. AS and MC also acknowledge
support by SECYT-UNC grant number PC No. 33620180101147CB. HC acknowledges
support from UTN grant UTI4915TC. Also, we thank the Centro de Cómputo de Alto
Desempeño (UNC), where the simulations were carried out. Authors acknowledge
use of data from STEREO (NASA) produced by the SECCHI consortium.

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# Quantum metrology with quantum Wheatstone bridge composed of Bose systems

Dong Xie<EMAIL_ADDRESS>School of Science, Guilin University of
Aerospace Technology, Guilin, Guangxi 541004, People’s Republic of China
Chunling Xu<EMAIL_ADDRESS>School of Science, Guilin University of
Aerospace Technology, Guilin, Guangxi 541004, People’s Republic of China
Xiwei Yao  School of Science, Guilin University of Aerospace Technology,
Guilin, Guangxi 541004, People’s Republic of China  An Min Wang  Department of
Modern Physics, University of Science and Technology of China, Hefei, Anhui
230026, People’s Republic of China

###### Abstract

The quantum version of a special classical Wheatstone bridge built with a
boundary-driven spin system has recently been proposed. We propose a quantum
Wheatstone bridge consisting of Bose systems, which can simulate the general
classical Wheatstone bridge. Unknown coupling can be obtained when the quantum
Wheatstone bridge is balanced, which can be determined simply by the homodyne
detection. When the expectation value of the homodyne detection is 0, the
quantum Wheatstone bridge is unbalanced. Regulate a known coupling strength to
make the expectation value of the homodyne detection be proportional to the
square root of the initial number of bosons, which means that the quantum
Wheatstone bridge is balanced. By calculating the quantum Fisher information,
we show that the measurement precision is optimal when the quantum Wheatstone
bridge is balanced. And the homodyne detection is close to the optimal
measurement in the case of low-temperature baths.

##  I Introduction

The classical Wheatstone bridge is an instrument for accurate measurement of
electrical resistance. Although its structure is simple, its accuracy and
sensitivity are relatively high, and it has been widely used in medical
diagnosis, testing instruments and automatic centering control lab1 ; lab2 ;
lab3 . It was invented by English inventor Christie in 1833 lab4 , but it
became known because Wheatstone lab5  was the first to use it to measure
resistance. As shown in Fig. 1, there are three known and tunable resistances
($R_{1},\ R_{2},\ R_{3}$), and the Resistance $R_{x}$ to be measured. By
varying the three tunable resistances until the indicator of galvanometer $A$
is 0, the classical Wheatstone bridge is in balance. At the balance point, the
value of $R_{x}$ can be obtained

$\displaystyle R_{x}=\frac{R_{2}}{R_{1}}R_{3}.$ (1)

With the study of smaller and smaller scale, quantum effects are gradually
prominent. Based on the quantum estimation theory lab6 , there are many works
exploring the use of quantum resources to improve parameter measurement
precision, such as, quantum magnetometer lab7 ; lab8 ; lab9 , quantum
thermometry lab10 ; lab11 ; lab12 ; lab13 ; lab14 ; lab15 , quantum
interferometry lab16 , and Quantum illumination lab17 ; lab18 ; lab19 .

Recently, Poulsen et al.lab20  utilized a few-body boundary-driven spin chain
to build a quantum version of the special Wheatstone bridge with
$R_{1}=R_{2}$. In this article, we propose a quantum Wheatstone bridge
consisting of Bose systems, which can simulate the general classical
Wheatstone bridge, as shown in Fig. 2. Similar to the classical Wheatstone
bridge, the unknown coupling strength $J_{x}$ in the quantum Wheatstone bridge
composed of Bose systems can be obtained at the balance point

$\displaystyle J_{x}=\frac{J_{2}}{J_{1}}J_{3}.$ (2)

In Ref. lab20 , they only considered the case of $J_{2}=J_{1}$. And they got
the unknown strength at the balance point by the formula $J_{x}=\lambda
J_{3}$, where the coefficient $\lambda$ was not equal to 1. It deviates from
the classical Wheatstone bridge and is therefore not a strictly quantum
Wheatstone bridge.

In this article, we are able to simply judge the bridge balance by the local
homodyne detection of cavity mode 2 or 3. When $J_{1}=J_{2}$, the quantum
Wheatstone bridge can be further simplified with $J_{0}=0$. By analytically
calculating the quantum Fisher information, we show that the homodyne
detection is close to the optimal measurement in the case of low-temperature
baths. The quantum fluctuation from the thermal baths will reduce the
measurement precision. With the optimal measurement, measurement precision
becomes more and more independent with the increase of fluctuation intensity.
For resisting the dissipation of cavity modes 2 and 3, the gain process is
proposed to maintain the criteria for quantum Wheatstone bridge balance.

Figure 1: Schematic diagram of the classical Wheatstone bridge. It is composed
of three controllable resistances ($R_{1},\ R_{2},\ R_{3}$), and the
Resistance $R_{x}$ to be measured. $A$ denotes a galvanometer. Figure 2:
Schematic diagram of the quantum Wheatstone bridge. This bridge is composed of
4 cavity modes. There are four tunable and known coupling strengths $J_{1},\
J_{2},\ J_{3},J_{0}$ and one coupling strength $J_{x}$ to be measured. Cavity
modes 1 and 4 are interacting with two thermal baths with temperature $T_{1}$
and $T_{4}$.

This article is organized as follows. In Section II, we introduce the setup of
the quantum Wheatstone bridge. In section III, the balance criterion of
quantum Wheatstone bridge is given. The measurement precision with the
homodyne detection is achieved in section IV. By calculating the quantum
Fisher information, the optimal precision is obtained in section V. We propose
a way to deal with the extra dissipation in section VI. The experiment
feasibility of our scheme is discussed in section VII. Finally, we make a
simple conclusion in section VIII.

##  II  setup consisting of the cavity modes

We propose to build the quantum Wheatstone bridge composed of Bose systems,
taking the cavity modes as an example, as shown in Fig. 2.

The total Hamiltonian can be described by ($\hbar=\kappa_{B}=1$ throughout
this article )

$\displaystyle H$
$\displaystyle=\sum_{i=1}^{4}\omega_{i}a^{\dagger}_{i}a_{i}+J_{1}a_{1}^{\dagger}a_{2}+J_{2}a_{1}^{\dagger}a_{3}+$
$\displaystyle
J_{3}a_{4}^{\dagger}a_{2}+J_{x}a_{4}^{\dagger}a_{3}+J_{0}a_{2}^{\dagger}a_{3}+h.c.,$
(3)

where $a_{i}$ ($a_{i}^{\dagger}$) denotes annihilation (creation) operator of
the $i$th cavity mode with the corresponding frequency $\omega_{i}$; $J_{1}$,
$J_{2}$, $J_{3}$, $J_{x}$, $J_{0}$ represent the coupling strength between
cavity modes 1 and 2, 1 and 3, 2 and 4, 3 and 4, 2 and 3, respectively. The
coupling strengths $J_{1},\ J_{2},\ J_{3},J_{0}$ are known and tunable. The
unknown coupling strength $J_{x}$ is what we want to measure.

The cavity modes 1 and 4 are coupled with two Markovian thermal baths. The
whole dynamic can be described by quantum Langevin-Heisenberg equation

$\displaystyle\dot{\vec{a}}=\mathbf{M}\vec{a}+\vec{a}_{\textmd{in}}$ (4)

where the cavity mode vector $\vec{a}=(a_{1},a_{2},a_{3},a_{4})^{\top}$, the
evolution matrix $\mathbf{M}$ is expressed as

$\mathbf{M}=\left(\begin{array}[]{ll}-i\omega_{1}-\kappa_{1}\ \ -iJ_{1}\ \
-iJ_{2}\ \ \ \ \ \ 0\\\ -iJ_{1}\ \ \ \ \ \ \ \ -i\omega_{2}\ \ -iJ_{0}\ \
-iJ_{3}\\\ -iJ_{2}\ \ \ \ \ \ \ \ -iJ_{0}\ \ -i\omega_{3}\ \ -iJ_{x}\\\ \ \ 0\
\ \ \ \ \ \ \ \ \ \ -iJ_{3}\ \ -iJ_{x}\ \
-i\omega_{4}-\kappa_{4}\end{array}\right).$

And
$\vec{a}_{\textmd{in}}=(\sqrt{2\kappa_{1}}a_{1\textmd{in}}(t),0,0,\sqrt{2\kappa_{4}}a_{4\textmd{in}}(t))^{T}$,
where the input noise operators $a_{j\textmd{in}}$ satisfy that

$\displaystyle\langle a_{j\textmd{in}}^{\dagger}(t)\rangle=0,$ (5)
$\displaystyle\langle
a_{j\textmd{in}}^{\dagger}(t)a_{j\textmd{in}}(t^{\prime})\rangle=N_{j}\delta(t-t^{\prime}),$
(6) $\displaystyle\langle
a_{j\textmd{in}}(t^{\prime})a_{j\textmd{in}}^{\dagger}(t)\rangle=(N_{j}+1)\delta(t-t^{\prime}),$
(7)

where the thermal average boson number
$N_{j}=\frac{1}{\exp(\omega_{j}/T_{j})-1}$ with $j=1,4$. $\kappa_{j}$ is the
coupling strength between the cavity mode $j$ and the thermal bath $j$ with
the temperature $T_{j}$. $\langle\mathbf{.}\rangle$ represents the mean over
the state of the bath degrees of freedom.

##  III  Quantum Wheatstone bridge balance criterion

When $J_{x}=\frac{J_{2}}{J_{1}}J_{3}$ at the balance point, the Hamiltonian in
Eq. (3) can be rewritten with bright mode $A_{+}=J_{1}a_{2}+J_{2}a_{3}$ and
dark mode $A_{-}=J_{2}a_{2}-J_{1}a_{3}$

$\displaystyle H$
$\displaystyle=\omega_{1}a^{\dagger}_{1}a_{1}+\omega_{4}a^{\dagger}_{4}a_{4}+\lambda_{+}A_{+}^{\dagger}A_{+}+\lambda_{-}A_{-}^{\dagger}A_{-}$
$\displaystyle a_{1}^{\dagger}A_{+}+J_{3}/J_{1}a_{4}^{\dagger}A_{+}+h.c.,$ (8)

The dark mode $A_{-}$ decouples with the cavity modes $a_{1}$ and $a_{4}$,
which are subjected to Markovian thermal baths. In order to make the
Hamiltonian in the above equation be equal to the Hamiltonian in Eq. (3). The
values of $\lambda_{+}$ and $\lambda_{-}$ should satisfy the following
conditions

$\displaystyle\lambda_{+}J_{1}^{2}+\lambda_{-}J_{2}^{2}=\omega_{2},$ (9)
$\displaystyle\lambda_{+}J_{2}^{2}+\lambda_{-}J_{1}^{2}=\omega_{3},$ (10)
$\displaystyle\lambda_{+}-\lambda_{-}=\frac{J_{0}}{J_{1}J_{2}}.$ (11)

The above equations have a solution only if

$\displaystyle\omega_{3}-\omega_{2}=J_{0}(\frac{J_{2}}{J_{1}}-\frac{J_{1}}{J_{2}}).$
(12)

When Eq. (2) and Eq. (12) hold, we can get the dark mode
$A_{-}(t)=J_{2}a_{2}(t)-J_{1}a_{3}(t)$, i.e.,

$\displaystyle A_{-}(t\rightarrow\infty)=e^{i\varphi(t)}A_{-}(t=0),$ (13)

where the phase $\varphi(t)$ is real. Based on the existence of the dark mode,
we next find the detail balance criterion of quantum Wheatstone bridge.

For $\kappa_{1},\kappa_{4}\gg|\omega_{i}-\omega_{j}|$ with
$\\{i,j\\}=\\{1,2,3,4\\}$, we can analytically obtain the solution of Eq. (4)
(see the detail in Appendix). The solutions of the modes $a_{2}(t)$ and
$a_{3}(t)$ can be given by

$\left(\begin{array}[]{ll}a_{2}(t)\\\
a_{3}(t)\end{array}\right)=e^{\mathbf{M}_{2}t}\left(\begin{array}[]{ll}{a_{2}(0)}\\\
{a_{3}(0)}\end{array}\right)+\int_{0}^{t}e^{\mathbf{M}_{2}(t-t^{\prime})}\left(\begin{array}[]{ll}{A_{2\textmd{in}}(t^{\prime})}\\\
{A_{3\textmd{in}}(t^{\prime})}\end{array}\right),$

where the evolution matrix $\mathbf{M}_{2}$ is described by

$\mathbf{M}_{2}=\left(\begin{array}[]{ll}-i\omega_{2}-\frac{J_{1}^{2}}{\kappa_{1}}-\frac{J_{3}^{2}}{\kappa_{4}}\
\ \ \ \ \ \
-iJ_{0}-\frac{J_{1}J_{2}}{\kappa_{1}}-\frac{J_{3}J_{x}}{\kappa_{4}}\\\
-iJ_{0}-\frac{J_{1}J_{2}}{\kappa_{1}}-\frac{J_{3}J_{x}}{\kappa_{4}}\ \
-i\omega_{3}-\frac{J_{2}^{2}}{\kappa_{1}}-\frac{J_{x}^{2}}{\kappa_{4}}\end{array}\right),$

and the noise operators are

$\displaystyle
A_{2\textmd{in}}(t)=-\sqrt{2}i[\frac{J_{1}a_{1\textmd{in}}(t)}{\sqrt{\kappa_{1}}}+\frac{J_{3}a_{4\textmd{in}}(t)}{\sqrt{\kappa_{4}}}],$
(16) $\displaystyle
A_{3\textmd{in}}(t)=-\sqrt{2}i[\frac{J_{2}a_{1\textmd{in}}(t)}{\sqrt{\kappa_{1}}}+\frac{J_{x}a_{4\textmd{in}}(t)}{\sqrt{\kappa_{4}}}].$
(17)

When $J_{1}\neq J_{2}$ and the quantum Wheatstone bridge is balanced, the
eigenvalues of $\mathbf{M}_{2}$ can be achieved

$\displaystyle E_{1}$
$\displaystyle=-\frac{i(J_{1}^{2}\omega_{2}-J_{2}^{2}\omega_{3})}{J_{1}^{2}-J_{2}^{2}},$
(18) $\displaystyle E_{2}$
$\displaystyle=-\frac{(J_{1}^{2}+J_{2}^{2})(J_{3}^{2}\kappa_{1}+J_{1}^{2}\kappa_{4})}{J_{1}^{2}\kappa_{1}\kappa_{4}}-\frac{i(J_{1}^{2}\omega_{2}-J_{2}^{2}\omega_{3})}{J_{1}^{2}-J_{2}^{2}}.$
(19)

The eigenvalue $E_{1}$ is pure imaginary, which means that
$e^{\mathbf{M}_{2}t}\neq\mathbf{0}$ for the long time limit
$t\rightarrow\infty$. It leads to that the expectations of the cavity modes 2
and 3 are described by

$\displaystyle\langle
a_{2}(t\rightarrow\infty)\rangle=\frac{e^{E_{1}t}J_{2}}{J_{1}^{2}+J_{2}^{2}}\langle
J_{2}a_{2}(0)-J_{1}a_{3}(0)\rangle;$ (20) $\displaystyle\langle
a_{3}(t\rightarrow\infty)\rangle=\frac{e^{E_{1}t}J_{1}}{J_{1}^{2}+J_{2}^{2}}\langle
J_{1}a_{3}(0)-J_{2}a_{2}(0)\rangle,$ (21)

where $\langle\mathbf{\star}\rangle$ denotes the mean over the initial state
of the cavity modes. When $J_{x}\neq\frac{J_{2}}{J_{1}}J_{3}$, the real parts
of the eigenvalues of $\mathbf{M}_{2}$ are all negative. As a result, the
expectation values $\langle a_{2}(t\rightarrow\infty)\rangle=\langle
a_{3}(t\rightarrow\infty)\rangle=0$.

When $J_{1}=J_{2}$, the frequency $\omega_{2}$ must be equal to $\omega_{3}$
for achieving the balanced quantum Wheatstone bridge. In this case, the
coupling strength $J_{0}$ can be arbitrary value, which is different from the
case of $J_{1}\neq J_{2}$. The corresponding eigenvalues of $\mathbf{M}_{2}$
are

$\displaystyle E^{\prime}_{1}$ $\displaystyle=i(J_{0}-\omega_{3}),$ (22)
$\displaystyle E^{\prime}_{2}$
$\displaystyle=i(J_{0}-\omega_{3})-\frac{2J_{1}^{2}}{\kappa_{1}}-\frac{2J_{3}^{2}}{\kappa_{4}}.$
(23)

For the long time limit $t\rightarrow\infty$, we get the similar form as the
previous results

$\displaystyle\langle
a_{2}(t\rightarrow\infty)\rangle=\frac{e^{E^{\prime}_{1}t}}{2}\langle
a_{2}(0)-a_{3}(0)\rangle;$ (24) $\displaystyle\langle
a_{3}(t\rightarrow\infty)\rangle=\frac{e^{E^{\prime}_{1}t}}{2}\langle
a_{3}(0)-a_{2}(0)\rangle.$ (25)

From Eq. (18), we can derive that ${\lim}_{J_{1}\rightarrow
J_{2}}E_{1}=-i\omega_{3}$, which is different from $E^{\prime}_{1}$ unless
that $J_{0}$ is equal to 0. As a result, there is a discontinuity for the
phase of $\langle a_{2}(t\rightarrow\infty)\rangle$ and $\langle
a_{3}(t\rightarrow\infty)\rangle$.

The detail balance criterion of quantum Wheatstone bridge can be be summarized
as follows: With the initial state of cavity modes 2 and 3 in the coherent
state $|\alpha\rangle|0\rangle$, it is balanced when the expectation values of
$\langle a_{2}(t\rightarrow\infty)\rangle$ and $\langle
a_{3}(t\rightarrow\infty)\rangle$ are not equal to 0; otherwise it is not
balanced.

##  IV  local homodyne detection

By the homodyne detection of the cavity mode $a_{2}(t\rightarrow\infty)$ or
$a_{3}(t\rightarrow\infty)$, we can obtain the information of $J_{x}$. The
uncertainty of $J_{x}$ can be achieved by the error transfer formula

$\displaystyle\delta J_{x}=\frac{\sqrt{\langle X^{2}\rangle-\langle
X\rangle^{2}}}{|\frac{d\langle X\rangle}{dJ_{x}}|},$ (26)

where $X$ denotes the measurement operator.

###  IV.1  In the case of $J_{1}=J_{2}$

When $J_{1}=J_{2}$ and $\omega_{2}=\omega_{3}$, we can analytically obtain the
differential coefficients

$\displaystyle\lim_{J_{x}\rightarrow J_{3}}\frac{d}{dJ_{x}}\langle
a_{2}(t\rightarrow\infty)\rangle=\frac{J_{3}\kappa_{1}e^{it(J_{0}-\omega_{3})}\langle
a_{2}(0)\rangle}{2(J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4}+iJ_{0}\kappa_{1}\kappa_{4})},$
(27) $\displaystyle\lim_{J_{x}\rightarrow J_{3}}\frac{d}{dJ_{x}}\langle
a_{3}(t\rightarrow\infty)\rangle=\frac{-J_{3}\kappa_{1}e^{it(J_{0}-\omega_{3})}\langle
a_{3}(0)\rangle}{2(J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4}+iJ_{0}\kappa_{1}\kappa_{4})}.$
(28)

For the long evolution time
$t>\tau=1/(\frac{2J_{1}^{2}}{\kappa_{1}}+\frac{2J_{3}^{2}}{\kappa_{4}})$, we
can derive the taylor series expansion

$\displaystyle\langle a_{2}(t)\rangle=$
$\displaystyle\frac{1}{2}e^{E^{\prime}_{1}t-\Gamma
y^{2}t-o(y^{3})t}\langle(1+\Lambda y+o(y^{2}))a_{2}(0)$
$\displaystyle-(1+o(y^{2}))a_{3}(0)\rangle;$ (29) $\displaystyle\langle
a_{3}(t)\rangle=$ $\displaystyle-\frac{1}{2}e^{E^{\prime}_{1}t-\Gamma
y^{2}t-o(y^{3})t}\langle 1+o(y^{2})a_{2}(0)$ $\displaystyle-(1-\Lambda
y+o(y^{2}))a_{3}(0)\rangle,$ (30)

where the coefficients
$\Gamma=\frac{J_{2}^{2}+iJ_{0}\kappa_{1}}{2(J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4}+iJ_{0}\kappa_{1}\kappa_{4})}$,
$\Lambda=\frac{J_{3}\kappa_{1}}{J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4}+iJ_{0}\kappa_{1}\kappa_{4}}$
and the difference $y=J_{x}-J_{3}$. For larger difference, $y\geq 1$, the
expectation values of $\langle a_{2}(t)\rangle$ and $\langle a_{3}(t)\rangle$
go to 0 very quickly. Therefore, we consider that $y<1$. It leads to that
higher-order terms $o(y^{3})t$ and $o(y^{2})$ can be ignored.

We consider the homodyne detection with the local quadrature operator
$X_{\varphi_{j}}=a_{j}e^{-i\varphi_{j}}+a_{j}^{\dagger}e^{-i\varphi_{j}}$,
where $\varphi_{j}$ represents phase with $j=2,3$. The initial states of
cavity 2 and 3 are given by the classical coherent state
$|\alpha\rangle_{2}|0\rangle_{3}$, with the real value $\alpha$. By utilizing
Eq. (26), we can obtain the measurement precision of $J_{x}$ with the
quadrature operator $X_{\varphi_{2}}$

$\displaystyle\delta J_{x}=\frac{{\sqrt{1+f}}}{|Re[(\Lambda-2\Gamma
yt(1+\Lambda y))e^{i\varphi_{2}+E^{\prime}_{1}t-\Gamma y^{2}t}]\alpha|},$ (31)

where the quantum fluctuation
$f=\frac{J_{1}^{2}N_{1}\kappa_{4}+J_{3}^{2}N_{4}\kappa_{1}}{J_{1}^{2}\kappa_{4}+J_{3}^{2}\kappa_{1}}$.
From the above equation, one can find that $\delta J_{x}$ takes the minimum
value at $y=0$. It shows that the optimal precision of $J_{x}$ can be obtained
when the Wheatstone bridge is balanced:

$\displaystyle\delta
J_{x}(y=0)=\frac{\sqrt{1+f}{\sqrt{(J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4})^{2}+J_{0}^{2}\kappa_{1}^{2}\kappa_{4}^{2}}}}{J_{3}\kappa_{1}|\cos(\varphi_{2}+\theta+(J_{0}-\omega_{3})t)|\alpha},$
(32)

where the phase $\theta$ satisfies that $e^{i\theta}=\Lambda/|\Lambda|$. When
the phase satisfies that $\varphi_{2}+\theta+(J_{0}-\omega_{3})t=n\pi$( $n$
represents an integer), the optimal precision by the homodyne detection can be
expressed as

$\displaystyle\delta
J^{o}_{x}=\frac{\sqrt{1+f}\sqrt{(J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4})^{2}+J_{0}^{2}\kappa_{1}^{2}\kappa_{4}^{2}}}{J_{3}\kappa_{1}\alpha}.$
(33)

Let $J_{0}=0$, the precision can be further improved,

$\displaystyle\delta
J^{o}_{x}(J_{0}=0)=\frac{\sqrt{1+f}(J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4})}{J_{3}\kappa_{1}\alpha}.$
(34)

###  IV.2  In the case of $J_{1}\neq J_{2}$

In the case of $J_{1}\neq J_{2}$, we can get the derivative at the balance
point

$\displaystyle\lim_{J_{x}\rightarrow J_{3}}\frac{d}{dJ_{x}}\langle
a_{2}(t\rightarrow\infty)\rangle=\frac{2J_{1}^{3}J_{2}J_{3}\kappa_{1}e^{E_{1}t}\langle
a_{2}(0)\rangle}{\mu}$
$\displaystyle-\frac{(J_{1}^{2}-J_{2}^{2})J_{1}^{2}J_{3}\kappa_{1}e^{E_{1}t}\langle
a_{3}(0)\rangle}{\mu},$ (35) $\displaystyle\lim_{J_{x}\rightarrow
J_{3}}\frac{d}{dJ_{x}}\langle
a_{3}(t\rightarrow\infty)\rangle=-\frac{2J_{1}^{3}J_{2}J_{3}\kappa_{1}e^{E_{1}t}\langle
a_{3}(0)\rangle}{\mu}$
$\displaystyle-\frac{(J_{1}^{2}-J_{2}^{2})J_{1}^{2}J_{3}\kappa_{1}e^{E_{1}t}\langle
a_{2}(0)\rangle}{\mu},$ (36)

where
$\mu=(J_{1}^{2}+J_{2}^{2})^{2}(J_{3}^{2}\kappa_{1}+J_{1}^{2}\kappa_{4}+iJ_{0}J_{1}\kappa_{1}\kappa_{4}/J_{2})$.
The measurement precision of $J_{x}$ can be given by

$\displaystyle\delta
J_{x}=\frac{\sqrt{1+f_{c}}|\mu|}{4J_{1}^{3}J_{2}J_{3}\kappa_{1}\alpha|\cos(\varphi_{2}+\theta^{\prime}+\frac{J_{1}^{2}\omega_{2}-J_{2}^{2}\omega_{3}}{J_{1}^{2}-J_{2}^{2}}t)|},$
(37)

where the phase $\theta^{\prime}$ satisfies that
$e^{i\theta^{\prime}}=|\mu|/\mu$ and the quantum fluctuation
$f_{c}=\frac{2J_{1}^{2}N_{1}\kappa_{4}+2J_{3}^{2}N_{4}\kappa_{1}}{(J_{1}^{2}\kappa_{4}+J_{3}^{2}\kappa_{1})(J_{1}^{2}+J_{2}^{2})}$.
When the phase satisfies that $\varphi_{2}+\theta^{\prime}-iE_{1}t=n\pi$, the
optimal precision with the homodyne detection can be expressed as

$\displaystyle\delta
J^{o}_{x}=\frac{\sqrt{1+f_{c}}|\mu|}{4J_{1}^{3}J_{2}J_{3}\kappa_{1}\alpha}.$
(38)

Figure 3: A plot of the logarithm of the uncertainty of $J_{x}$ versus
$J_{x}$. Here, the dimensionless parameters are set as: $\omega_{2}=100$,
$\omega_{3}=101$, $\alpha=10000$, $\kappa_{1}=10$, $\kappa_{4}=10$,
$J_{1}=10$,$J_{2}=15$,$J_{3}=10$, and
$J_{0}=(\omega_{3}-\omega_{2})J_{1}J_{2}/(J_{2}^{2}-J_{1}^{2})=1.2$. At the
balance point, the unknown coupling strength $J_{x}=J_{2}J_{3}/J_{1}=15$.

When $J_{1}=J_{2}$, Eq. (38) becomes Eq. (33). It shows that the optimal
precision is still continuous at the point $J_{1}=J_{2}$ and not affected by
the discontinuity for the phase of $\langle a_{2}(t\rightarrow\infty)\rangle$
and $\langle a_{3}(t\rightarrow\infty)\rangle$. This is mainly because the
measurement angle $\varphi_{2}$ can compensate of the effect of phase.

As shown in Fig. 3, we can see that the minimum uncertainty of $J_{x}$ appears
when the quantum Wheatstone bridge is balanced, i.e.,
$J_{x}=J_{2}J_{3}/J_{1}=15$. Consistent with the previous results, the
measurement precision at the balance point is optimal.

##  V Quantum Fisher information of $J_{x}$

The quantum Cramér-Rao bound lab21 ; lab22 ; lab23 ; lab24  provides the bound
on the estimation precision of parameter $J_{x}$,

$\displaystyle\delta J_{x}\geq\frac{1}{\mathcal{F}[J_{x}]},$ (39)

where $\mathcal{F}[J_{x}]$ denotes the quantum Fisher information of $J_{x}$,
which is encoded into the cavity modes. Due to that Hamiltonian is quadratic
and the initial state is Gaussian, the steady state of cavity modes 2 and 3 is
Gaussian too lab24a . The Quantum Fisher information for Gaussian state is
obtained through the fidelity by Pinel et al. in 2013 lab25 ,

$\displaystyle\mathcal{F}[J_{x}]=\frac{1}{2(1+P_{J_{x}}^{2})}\textmd{Tr}[(\mathcal{C}^{-1}_{J_{x}}\mathcal{C}^{{}^{\prime}}_{J_{x}})^{2}]+\frac{2P_{J_{x}}^{\prime
2}}{1-P_{J_{x}}^{4}}$
$\displaystyle+{\langle\mathbf{X}^{\top}\rangle}^{\prime}_{J_{x}}\mathcal{C}^{-1}_{J_{x}}\langle\mathbf{X}\rangle^{\prime}_{J_{x}},$
(40)

where $P_{J_{x}}=\frac{1}{2d}$, $d=\sqrt{\textmd{Det }\mathcal{C}_{J_{x}}}$
and $A^{\prime}_{J_{x}}$ is the term by term derivative of $A_{J_{x}}$ with
respect to ${J_{x}}$. The entries of the covariance matrix $\mathcal{C}$ are
defined as
$\mathcal{C}_{ij}:=\frac{1}{2}\langle\\{\mathbf{X}_{i},\mathbf{X}_{j}\\}\rangle-\langle\mathbf{X}_{i}\rangle\langle\mathbf{X}_{j}\rangle$,
with $\langle\bullet\rangle$ the expectation value. The vector of quadrature
operators is $\mathbf{X}=(q_{2},q_{3},p_{2},p_{3})^{\top}$, where the
quadrature operators are defined as $q_{j}:=(a_{j}+a^{\dagger}_{j})$ and
$p_{j}:=\frac{1}{i}(a_{j}-a^{\dagger}_{j})$.

By utilizing Eq. (20) and Eq. (21), we can obtain the covariance matrix

$\mathbf{\mathcal{C}}=\left(\begin{array}[]{ll}1+f_{c}\ \ \ \
\frac{f_{c}J_{2}}{J_{1}}\ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ 0\\\
\frac{f_{c}J_{2}}{J_{1}}\ \ \ \ 1+\frac{f_{c}J_{2}^{2}}{J_{1}^{2}}\ \ \ \ \ \
0\ \ \ \ \ \ \ \ \ \ 0\\\ \ \ 0\ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ 1+f_{c}\ \
\ \ \ \frac{f_{c}J_{2}}{J_{1}}\\\ \ \ 0\ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \
\frac{f_{c}J_{2}}{J_{1}}\ \ \ \ \ \
1+\frac{f_{c}J_{2}^{2}}{J_{1}^{2}}\end{array}\right).$

And using Eq. (35) and Eq. (36), the derivative of the vector of quadrature
operators is given by

$\displaystyle\langle\mathbf{X}\rangle^{\prime}_{J_{x}}=\frac{\alpha
J_{1}^{2}J_{3}\kappa_{1}}{|\mu|}(4J_{1}J_{2}\cos\phi,2(J_{2}^{2}-J_{1}^{2})\cos\phi,$
$\displaystyle 4J_{1}J_{2}\sin\phi,2(J_{2}^{2}-J_{1}^{2})\sin\phi),$ (42)

where $\phi=\theta^{\prime}-iE_{1}t$.

When the input probes are sufficiently strong, the quantum Fisher information
can be dominated by the last term in Eq. (40)lab26

$\displaystyle\mathcal{F}[J_{x}]\approx{\langle\mathbf{X}^{\top}\rangle}^{\prime}_{J_{x}}\mathcal{C}^{-1}_{J_{x}}\langle\mathbf{X}\rangle^{\prime}_{J_{x}}$
(43) $\displaystyle=\frac{(\alpha
J_{1}^{2}J_{3}\kappa_{1})^{2}}{|\mu|^{2}}\frac{4(1+f_{c})J_{1}^{2}(J_{1}^{2}+J_{2}^{2})^{2}}{(1+f_{c})J_{1}^{2}+f_{c}J_{2}^{2}}$
(44)

According to the quantum Cramér-Rao bound, the uncertainty of $J_{x}$ is

$\displaystyle\delta
J_{x}\geq\frac{g\sqrt{1+f_{c}}|\mu|}{4J_{1}^{3}J_{2}J_{3}\kappa_{1}\alpha},$
(45)

where the coefficient
$g=\frac{2J_{2}\sqrt{(1+f_{c})J_{1}^{2}+f_{c}J_{2}^{2}}}{(J_{1}^{2}+J_{2}^{2})(1+f_{c})}$.

By a simple calculation, we can prove that

$\displaystyle
1-g^{2}=\frac{[(1+f_{c})J_{1}^{2}+(f_{c}-1)J_{2}^{2}]^{2}}{(J_{1}^{2}+J_{2}^{2})^{2}(1+f_{c})^{2}}\geq
0.$ (46)

It means that the coefficient $g\leq 1$. When
$J_{2}^{2}=\frac{1+f_{c}}{1-f_{c}}J_{1}^{2}$, the equality holds, i.e., $g=1$.
Comparing with Eq. (38), it shows that the homodyne detection is the optimal
measurement in the case of $J_{2}^{2}=\frac{1+f_{c}}{1-f_{c}}J_{1}^{2}$ and
$\alpha\gg 1$. In order to get the equation
$J_{2}^{2}=\frac{1+f_{c}}{1-f_{c}}J_{1}^{2}$, the quantum fluctuation $f_{c}$
must be less than 1. It means that the temperature of both thermal baths
should be relatively small. When $f_{c}=0$, i.e., the temperature of both
thermal baths is 0 ($T_{1}=T_{4}=0$)

$\displaystyle\delta
J_{x}\geq\frac{|\mu|}{2J_{1}^{2}J_{3}\kappa_{1}{(J_{1}^{2}+J_{2}^{2})}\alpha}.$
(47)

For large quantum fluctuation $f_{c}>1$, the homodyne detection is not the
optimal measurement. With the homodye detection, the precision of $J_{x}$
decreases as the quantum fluctuation term $f_{c}$ increases. With the optimal
measurement, the precision of $J_{x}$ becomes increasingly independent of the
quantum fluctuation term $f_{c}$. For $f_{c}\gg 1$, the precision of $J_{x}$
is given by

$\displaystyle\delta
J_{x}\geq\frac{|\mu|}{2J_{1}^{3}J_{3}\kappa_{1}\sqrt{J_{1}^{2}+J_{2}^{2}}\alpha}.$
(48)

Especially, in the case of $J_{1}=J_{2}$, $\omega_{2}=\omega_{3}$ and
$J_{0}=0$, the optimal measurement precision of $J_{x}$ is given by

$\displaystyle\delta
J_{x}^{o}=\frac{\sqrt{1+2f}(J_{3}^{2}\kappa_{1}+J_{2}^{2}\kappa_{4})}{J_{3}\kappa_{1}\alpha\sqrt{1+f}}.$
(49)

From the above equation, we can see that the measurement uncertainty of
$J_{x}$ increases with the quantum fluctuation $f_{c}$. And the ratio $\delta
J_{x}(f_{c}=0)/\delta J_{x}(f_{c}\rightarrow\infty)=1/\sqrt{2}$. It shows that
quantum fluctuations can reduce the precision by up to a factor of
$1/\sqrt{2}$. However, the uncertainty with the local homodyne detection still
increases. Hence, it is necessary to construct an optimal nonlocal measurement
operator to reduce the influence of the quantum fluctuation.

##  VI  Dissipation of cavity 2 and 3

When the cavity modes 2 and 3 are not perfect, there are intrinsic losses.
This will prevent us from determining whether quantum Wheatstone bridge is
balanced. This can be overcome by using gains to make up for losses. The gain
process can be achieved by coupling with an amplifying channel lab26  or
harnessing the feeding fields lab27 ; lab28 .

Including the intrinsic losses and the gains, quantum Langevin-Heisenberg
equation of the cavity modes 2 and 3 can be described as

$\displaystyle\dot{a_{2}}=-iJ_{1}a_{1}+(-i\omega_{2}-\kappa_{2}+\gamma_{2})a_{2}-iJ_{0}a_{3}-iJ_{3}a_{4}+$
$\displaystyle\sqrt{2k_{2}}a_{2\textmd{in}}(t)-\sqrt{2\gamma_{2}}d_{2\textmd{in}}^{\dagger}(t);$
(50)
$\displaystyle\dot{a_{3}}=-iJ_{2}a_{1}-iJ_{0}a_{2}+(-i\omega_{3}-\kappa_{3}+\gamma_{3})a_{3}-iJ_{x}a_{4}+$
$\displaystyle\sqrt{2k_{3}}a_{3\textmd{in}}(t)-\sqrt{2\gamma_{3}}d_{3\textmd{in}}^{\dagger}(t);$
(51)

where $\kappa_{2}$, $\kappa_{3}$ are the intrinsic loss rates of the cavity
modes 2, 3, respectively; $\gamma_{2}$, $\gamma_{3}$ are the gain rates of the
cavity modes 2,3 respectively; the noise operator $Y_{j\textmd{in}}(t)$
satisfies

$\displaystyle\langle Y_{j\textmd{{in}}}^{\dagger}(t)\rangle=0,$ (52)
$\displaystyle\langle
Y_{j\textmd{{in}}}^{\dagger}(t)Y_{j\textmd{in}}(t^{\prime})\rangle=0,$ (53)
$\displaystyle\langle
Y_{j\textmd{{in}}}(t^{\prime})Y_{j\textmd{in}}^{\dagger}(t)\rangle=\delta(t-t^{\prime}),$
(54)

where $Y={a,d}$ and $j=2,3$.

Let the gain rate $\gamma_{j}$ be equal to the corresponding loss rate
$\kappa_{j}$($\kappa_{j}=\gamma_{j}$), the quantum Wheatstone bridge balance
criterion is reestablished. In the case of $\kappa_{j}=\gamma_{j}$, the
solution can be obtained

$\left(\begin{array}[]{ll}a_{2}(t)\\\
a_{3}(t)\end{array}\right)=e^{\mathbf{M}_{2}t}\left(\begin{array}[]{ll}{a_{2}(0)}\\\
{a_{3}(0)}\end{array}\right)+\int_{0}^{t}e^{\mathbf{M}_{2}(t-t^{\prime})}\left(\begin{array}[]{ll}{A_{2\textmd{in}}(t^{\prime})}\\\
{A_{3\textmd{in}}(t^{\prime})}\end{array}\right),$

where the noise operators are redefined as

$\displaystyle
A_{2\textmd{in}}(t)=-\sqrt{2}i[\frac{J_{1}a_{1\textmd{in}}(t)}{\sqrt{\kappa_{1}}}+\frac{J_{3}a_{4\textmd{in}}(t)}{\sqrt{\kappa_{4}}}]+$
$\displaystyle\sqrt{2k_{2}}(a_{2\textmd{in}}(t)-d_{2\textmd{in}}^{\dagger}(t)),$
(56) $\displaystyle
A_{3\textmd{in}}(t)=-\sqrt{2}i[\frac{J_{2}a_{1\textmd{in}}(t)}{\sqrt{\kappa_{1}}}+\frac{J_{x}a_{4\textmd{in}}(t)}{\sqrt{\kappa_{4}}}]+$
$\displaystyle\sqrt{2k_{3}}(a_{3\textmd{in}}(t)-d_{3\textmd{in}}^{\dagger}(t)).$
(57)

For simplicity, we assume $\kappa_{2}=\kappa_{3}$. As a result, the optimal
measurement precision can be described as

$\displaystyle\delta
J_{x}\geq\frac{g\sqrt{1+f^{\prime}_{c}}|\mu|}{4J_{1}^{3}J_{2}J_{3}\kappa_{1}\alpha},$
(58)

where the quantum fluctuation
$f^{\prime}_{c}=f_{c}+\frac{\kappa_{1}\kappa_{2}\kappa_{4}}{2(J_{1}^{2}\kappa_{4}+J_{3}^{2}\kappa_{1})}$.
The increment of quantum fluctuation $f^{\prime}_{c}-f_{c}$ is from the
dissipation and the gain environment of the cavity modes 2 and 3. The gain can
compensate for the dissipation and reach the balanced quantum Wheatstone
bridge, but it can not completely eliminate the effect of quantum fluctuation
from the dissipative cavity modes 2 and 3 on the estimation precision. For the
quantum fluctuation $f_{c}\gg 1$, the estimation precision is independent of
$f_{c}$. In this case, the dissipation and the gain environment of the cavity
modes 2 and 3 do not result in a loss of the estimation precision.

##  VII Possible experimental implementations

Our model with $J_{0}=0$ can be realized in the opto-mechanical system. Let
the cavity modes 2 and 3 become two mechanical modes. The total Hamiltonian
can be rewritten as

$\displaystyle H_{\textmd{om}}$
$\displaystyle=\sum_{i=1}^{4}\omega_{i}a^{\dagger}_{i}a_{i}+G_{1}a_{1}^{\dagger}a_{1}(a_{2}+a_{2}^{\dagger})+G_{2}a_{1}^{\dagger}a_{1}(a_{3}+a_{3}^{\dagger})+$
$\displaystyle
G_{3}a_{4}^{\dagger}a_{4}(a_{2}+a_{2}^{\dagger})+G_{x}a_{4}^{\dagger}a_{4}(a_{3}+a_{3}^{\dagger}),$
(59)

where $a_{2}$ and $a_{3}$ represent the bosonic annihilation operators of the
mechanical modes. Replacing the cavity modes with the sum of its quantum
fluctuation operator classical mean value, the Hamiltonian can be linearized
in the interaction picture as lab29

$\displaystyle
H_{\textmd{om,int}}=J_{1}a_{1}^{\dagger}a_{2}+J_{2}a_{1}^{\dagger}a_{3}+J_{3}a_{4}^{\dagger}a_{2}+J_{x}a_{4}^{\dagger}a_{3}+h.c.,$
(60)

where $J_{i}=G_{i}\beta_{i}$ is the effective optomechanical coupling strength
and classical amplitude $\beta_{i}$ can be controlled by the classical driving
with $i=(1,\ 2,\ 3,\ x)$.

In experiment, this setup has been created by a superconducting circuit of
aluminum on a sapphire substrate lab30 , the microchip circulator device
composed of three high-impedance spiral inductors capacitively coupled to the
in-plane vibrational modes of a dielectric nanostring mechanical resonator
lab31 . Therefore, our proposed scheme can be implemented using current
experimental technology.

##  VIII Conclusion

We utilized Bose system to construct the general quantum Wheatstone bridge. We
propose a more feasible criterion for determining the balance of the quantum
Wheatstone bridge. By calculating the quantum Fisher information, we show that
the local homodyne detection is close to the optimal measurement with low
temperature of the Markovian baths. With the optimal measurement, the
measurement precision of the unknown coupling strength is independent of the
quantum fluctuation from the baths with high temperature. In order to maintain
the balance criterion of the quantum Wheatstone bridge, extra gain process is
proposed to resist the dissipation. Our proposed scheme can be realized in the
current optomechanical systems, which will facilitate the precision
measurement of photomechanical coupling strength.

## Acknowledgements

This research was supported by the National Natural Science Foundation of
China (Grant No. 62001134), Guangxi Natural Science Foundation (Grant
No.2020GXNSFAA159047, 2020GXNSFAA159007), and National Key R&D Program of
China (Grant No. 2018YFB1601402-2).

## Appendix A

Using the slowly varying operators $S_{i}=a_{i}e^{i\omega_{i}t}$, the Eq.(A2)
can be rewritten as

$\displaystyle\dot{\vec{S}}=\mathbf{M}\vec{S}+\vec{a^{\prime}}_{in}$ (A1)

where
$\vec{a^{\prime}}_{in}=(e^{i\omega_{1}t}\sqrt{2\kappa_{1}}a_{1in}(t),0,0,e^{i\omega_{4}t}\sqrt{2\kappa_{4}}a_{4in}(t))^{T}$,
and the evolution matrix $\mathbf{M}$ is expressed as

$\mathbf{M^{\prime}}=\left(\begin{array}[]{ll}-\kappa_{1}\ \ -iJ^{\prime}_{1}\
\ -iJ^{\prime}_{2}\ \ \ \ \ \ 0\\\ -iJ^{\prime*}_{1}\ \ \ \ 0\ \ \ \
-iJ^{\prime}_{0}\ \ -iJ^{\prime}_{3}\\\ -iJ^{\prime*}_{2}\ \
-iJ^{\prime*}_{0}\ \ \ 0\ \ \ \ -iJ^{\prime}_{x}\\\ \ \ 0\ \ \ \
-iJ^{\prime*}_{3}\ \ -iJ^{\prime*}_{x}\ \ -\kappa_{4}\end{array}\right),$

where $J^{\prime}_{1}=J_{1}e^{-i\Delta_{21}t}$,
$J^{\prime}_{2}=J_{2}e^{-i\Delta_{31}t}$,
$J^{\prime}_{3}=J_{3}e^{-i\Delta_{42}t}$,
$J^{\prime}_{x}=J_{x}e^{-i\Delta_{43}t}$ and
$J^{\prime}_{0}=J_{0}e^{-i\Delta_{32}t}$ with the frequency detunings between
two cavity modes $\Delta_{ij}=\omega_{i}-\omega_{j}$.

We can obtain the formal solutions of $S_{1}$ and $S_{4}$

$\displaystyle
S_{1}=-i\int_{0}^{t}dt^{\prime}e^{-\kappa_{1}(t-t^{\prime})}[J^{\prime}_{1}S_{2}(t^{\prime})+J^{\prime}_{2}S_{3}(t^{\prime})]$
$\displaystyle+\int_{0}^{t}dt^{\prime}e^{-\kappa_{1}(t-t^{\prime})}[\sqrt{2\kappa_{1}}a^{\prime}_{1in}(t^{\prime})];$
(A3) $\displaystyle
S_{4}=-i\int_{0}^{t}dt^{\prime}e^{-\kappa_{4}(t-t^{\prime})}[J^{\prime*}_{3}S_{2}(t^{\prime})+J^{\prime*}_{x}S_{3}(t^{\prime})]$
$\displaystyle+\int_{0}^{t}dt^{\prime}e^{-\kappa_{4}(t-t^{\prime})}[\sqrt{2\kappa_{4}}a^{\prime}_{4in}(t^{\prime})].$
(A4)

We consider that the dissipation rates of the cavity modes $a_{1}$ and $a_{4}$
is much larger than the cases of the cavity modes $a_{2}$ and $a_{3}$, i.e.,
$\kappa_{1}$,$\kappa_{4}\gg\kappa_{2}$,$\kappa_{3}$, where we assume that the
dissipation rates of the cavity modes $a_{2}$ and $a_{3}$:
$\kappa_{2}=0$,$\kappa_{3}=0$ for simplicity. The changes of mode $S_{2}$ and
$S_{3}$ are small within the range of the integration of the cavity modes
$S_{1}$ and $S_{4}$. Therefore, we can set $S_{2}(t^{\prime})=S_{2}(t)$ and
$S_{3}(t^{\prime})=S_{3}(t)$ in Eq. (A3) and Eq. (A4) to obtain

$\displaystyle
S_{1}=\frac{-iJ_{1}S_{2}}{\kappa_{1}-i\Delta_{21}}e^{-i\Delta_{21}t}+\frac{-iJ_{2}S_{3}}{\kappa_{1}-i\Delta_{31}}e^{-i\Delta_{31}t}$
$\displaystyle+\frac{\sqrt{2\kappa_{1}}a_{1\textmd{in}}}{\kappa_{1}}e^{-i\omega_{1}t};$
(A5) $\displaystyle
S_{4}=\frac{-iJ_{3}S_{2}}{\kappa_{4}-i\Delta_{24}}e^{-i\Delta_{24}t}+\frac{-iJ_{x}S_{3}}{\kappa_{4}-i\Delta_{34}}e^{-i\Delta_{34}t}$
$\displaystyle+\frac{\sqrt{2\kappa_{4}}a_{4\textmd{in}}}{\kappa_{4}}e^{-i\omega_{4}t}.$
(A6)

Substituting the above equations into Eq. (A1), the modes $S_{1}$ and $S_{4}$
can be adiabatically eliminated. The corresponding evolution equations of the
modes $S_{2}$ and $S_{3}$ are reduced to

$\displaystyle\dot{S_{2}}=(-\frac{J_{1}^{2}}{\kappa_{1}-i\Delta_{21}}-\frac{J_{3}^{2}}{\kappa_{1}-i\Delta_{24}})S_{2}+$
$\displaystyle(-\frac{J_{1}J_{2}}{\kappa_{1}-i\Delta_{31}}-\frac{J_{3}J_{x}}{\kappa_{4}-i\Delta_{34}}-iJ_{0})e^{-i\Delta_{32}t}S_{3}-$
$\displaystyle(\frac{J_{1}a_{1\textmd{in}}}{\sqrt{\kappa_{1}}}+\frac{J_{3}a_{4\textmd{in}}}{\sqrt{\kappa_{4}}})\sqrt{2}ie^{-i\omega_{2}t};$
(A7)
$\displaystyle\dot{S_{3}}=(-\frac{J_{1}J_{2}}{\kappa_{1}-i\Delta_{21}}-\frac{J_{3}J_{x}}{\kappa_{1}-i\Delta_{24}}-iJ_{0})e^{-i\Delta_{23}t}S_{2}+$
$\displaystyle(\frac{J_{2}^{2}}{\kappa_{1}-i\Delta_{31}}-\frac{J_{x}^{2}}{\kappa_{4}-i\Delta_{34}})S_{3}-$
$\displaystyle(\frac{J_{2}a_{1\textmd{in}}}{\sqrt{\kappa_{1}}}+\frac{J_{x}a_{4\textmd{in}}}{\sqrt{\kappa_{4}}})\sqrt{2}ie^{-i\omega_{3}t}.$
(A8)

For $\kappa_{1},\kappa_{4}\gg\Delta_{ij}$, combine these equations with
$S_{2}=a_{2}e^{i\omega_{2}t}$ and $S_{3}=a_{3}e^{i\omega_{3}t}$, we can obtain

$\left(\begin{array}[]{ll}\dot{a}_{2}(t)\\\
\dot{a}_{3}(t)\end{array}\right)=\mathbf{M}_{2}\left(\begin{array}[]{ll}{a_{2}(t)}\\\
{a_{3}(t)}\end{array}\right)+\left(\begin{array}[]{ll}{A_{2\textmd{in}}(t)}\\\
{A_{3\textmd{in}}(t)}\end{array}\right),$

where the evolution matrix $\mathbf{M}_{2}$ is described by

$\mathbf{M}_{2}=\left(\begin{array}[]{ll}-i\omega_{2}-\frac{J_{1}^{2}}{\kappa_{1}}-\frac{J_{3}^{2}}{\kappa_{4}}\
\ \ \ \ \ \
-iJ_{0}-\frac{J_{1}J_{2}}{\kappa_{1}}-\frac{J_{3}J_{x}}{\kappa_{4}}\\\
-iJ_{0}-\frac{J_{1}J_{2}}{\kappa_{1}}-\frac{J_{3}J_{x}}{\kappa_{4}}\ \
-i\omega_{3}-\frac{J_{2}^{2}}{\kappa_{1}}-\frac{J_{x}^{2}}{\kappa_{4}}\end{array}\right),$

and the noise operators are

$\displaystyle
A_{2in}(t)=-\sqrt{2}i[\frac{J_{1}a_{1\textmd{in}}(t)}{\sqrt{\kappa_{1}}}+\frac{J_{3}a_{4\textmd{in}}(t)}{\sqrt{\kappa_{4}}}],$
(A11) $\displaystyle
A_{3in}(t)=-\sqrt{2}i[\frac{J_{2}a_{1\textmd{in}}(t)}{\sqrt{\kappa_{1}}}+\frac{J_{x}a_{4\textmd{in}}(t)}{\sqrt{\kappa_{4}}}].$
(A12)

The solutions of the modes $a_{2}(t)$ and $a_{3}(t)$ can be obtained

$\left(\begin{array}[]{ll}a_{2}(t)\\\
a_{3}(t)\end{array}\right)=e^{\mathbf{M}_{2}t}\left(\begin{array}[]{ll}{a_{2}(0)}\\\
{a_{3}(0)}\end{array}\right)+\int_{0}^{t}e^{\mathbf{M}_{2}(t-t^{\prime})}\left(\begin{array}[]{ll}{A_{2\textmd{in}}(t^{\prime})}\\\
{A_{3\textmd{in}}(t^{\prime})}\end{array}\right).$

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