# Phonon-laser ultrasensitive force sensor

Zhichao Liu  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation
Academy of Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  University of the Chinese Academy of Sciences,
Beijing 100049, China  Yaqi Wei  State Key Laboratory of Magnetic Resonance
and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,
Innovation Academy of Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  University of the Chinese Academy of
Sciences, Beijing 100049, China  Liang Chen  Corresponding author:
<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation
Academy of Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  Research Center for Quantum Precision
Measurement, Guangzhou Institute of Industry Technology, Guangzhou, 511458,
China  Ji Li  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation
Academy of Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  University of the Chinese Academy of Sciences,
Beijing 100049, China  Shuangqing Dai  State Key Laboratory of Magnetic
Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and
Mathematics, Innovation Academy of Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  University of
the Chinese Academy of Sciences, Beijing 100049, China  Fei Zhou  State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan
Institute of Physics and Mathematics, Innovation Academy of Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,
China  Research Center for Quantum Precision Measurement, Guangzhou Institute
of Industry Technology, Guangzhou, 511458, China  Mang Feng  Corresponding
author<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,
Innovation Academy of Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  University of the Chinese Academy of
Sciences, Beijing 100049, China  Research Center for Quantum Precision
Measurement, Guangzhou Institute of Industry Technology, Guangzhou, 511458,
China  School of Physics, Zhengzhou University, Zhengzhou 450001, China

###### Abstract

Developing nano-mechanical oscillators for ultrasensitive force detection is
of great importance in exploring science. We report our achievement of
ultrasensitive detection of the external force regarding the radio-frequency
electric field by a nano-sensor made of a single trapped 40Ca+ ion under
injection-locking, where squeezing is additionally applied to detection of the
smallest force in the ion trap. The employed ion is confined stably in a
surface electrode trap and works as a phonon laser that is very sensitive to
the external disturbance. The injection-locking drove the ion’s oscillation
with phase synchronization, yielding the force detection with sensitivity of
347 $\pm$ 50 yN/$\sqrt{Hz}$. Further with 3 dB squeezing applied on the
oscillation phase variance, we achieved a successful detection of the smallest
force to be 86.5 $\pm$ 70.1 yN.

##  I INTRODUCTION

With development of nanotechnology using atomic and molecular sensors, the
physical quantities have been detected in a higher level of sensitivity than
ever before due to the excellent perceived performance of the sensors
Hosten2016 ; Schreppler2019 ; Mamin2001 ; Lecocq2015 ; Caves1980 ; Degen2017 ;
Kotler12011 ; Kim2017 . Force detection has been always taking an important
role in precision measurement since all the changes of motion are concerned
with the force due to Newton’s laws, where the small force detection by
trapped ions or atoms has recently demonstrated the superiority of high
sensitivity and broad adjustability Gilmore2017 ; Affolter2020 ; Gilmore2021 ;
Shaniv2017 ; Biercuk2010 ; Blums2018 ; Araneda2019 ; Knunz2010 . The weak AC
electric force was sensitively detected by means of spin-motion entanglement
in two-dimensional trapped-ion mechanical oscillators Gilmore2017 ;
Affolter2020 ; Gilmore2021 . The radio-frequency electric force as small as 5
yN was measured, based on the technique of injection-locked phonon laser, by
observing the range of injected frequency and the gravity variance of the
ion’s displacement Knunz2010 . Such a radio-frequency electric force was also
detected by phase-coherent Doppler velocimetry, reaching the sensitivity of
390 $\pm$ 150 yN/$\sqrt{Hz}$ Biercuk2010 . A recent detection of the sub-
attonewton force regarding DC electric field was accomplished by measuring the
ion’s displacement three-dimensionally at nanometer precision Blums2018 .

In the present work, we report our ultrasensitive detection of the radio-
frequency electric force at hundred yoctonewton’s order of magnitude by a
nano-probe made of a single trapped 40Ca+ ion. Although our detection, similar
to in Knunz2010 , also takes advantage of the stably confined ion behaving as
a phonon laser, we acquired the sensitivity of the sensor and the lower bound
of the small force by faster measurements compared to the method of scanning
the injection frequency in Knunz2010 . In addition, squeezing technique
Natarajan1995  was employed to further suppress thermal noises in the system.
The ion’s phonon laser under our consideration works as an amplitude-amplified
harmonic oscillator, which is quite sensitive to the external force, as
reported previously Vahala2009 ; Grudinin2010 ; Ip2018 ; He2016 ; Hush2015 ;
Dominguez2017 . In our experiment, the trapped 40Ca+ ion was first Doppler-
cooled and then pumped simultaneously by a red-detuned laser and a blue-
detuned laser with respect to the resonant transition. After locking the
oscillation frequency by injection signal, we might detect the injected force,
which is induced by a radio-frequency electric field, based on the photon
count measurements of the oscillation amplitude, the latter of which was
carried out by fitting the fluorescence curve directly acquired from the
photomultiplier tube (PMT) along with the synchronized measurement of the
injection-locking signal. Our achieved force sensitivity is as low as 347
$\pm$ 50 yN/$\sqrt{Hz}$, and the oscillation amplitude uncertainty is only 15
nm with respect to the oscillation amplitude $\approx$18 $\mu$m. The detection
sensitivity represents the minimum detectable change rate of the force. In
contrast, we may also try to detect the smallest force, indicating the minimum
force our probe could sense. To this end, we introduce the squeezing technique
as in Natarajan1995  and with 3 dB squeezing applied on the oscillation phase
variance, we detected successfully the smallest force of 86.5 $\pm$ 70.1 yN.
Our technique could be applied to the oscillation force sensors using other
ions, atoms or solid-state oscillators.

Figure 1: Experimental system and scheme. (a) Sketch of the phonon laser of an
ion trapped above the surface of the SET. The 40Ca+ ion with oscillating
motion irradiated by red- and blue-detuned 397-nm laser beams is trapped by
the potential $\Psi(t)$ regarding a time-varying electric field. The
electrodes are labelled with names for convenience of description in the text,
where EE, SE, ME, AE and RF indicate, respectively, the endcap electrode, the
steering electrode, the middle electrode, the axial electrode and the radio-
frequency electrode. The injection signal applied on AE with frequency
$\omega_{i}=\omega_{z}$, locking the oscillation frequency, is to be detected.
The squeezing, with frequency doubling of $\omega_{i}$, acts on EEs when
required. The top-right inset is the level scheme of 40Ca+ for our purpose.
(b) Signal processing scheme of the photon-arrival-time measurements. The
applied injection signal and the squeezing signal are phase synchronized, with
the latter of frequency doubling with respect to the former. The photon-
arrival-time transited to voltage is measured by TAC. The ratio of the beam
splitter for the PMT to CCD is set to be 7/3. (c) Signal and sequence diagram.
The Injection signal contributes to locking the oscillation frequency/phase.
The Squeezing signal is applied, when required, to suppress the variance of
the oscillation phase. The Gate signal controls the measurement time $t_{m}$
containing a series of oscillation period $T$ and the TAC records the photon’s
arrival time $t$ in each period. The Start signal comes from the PMT regarding
the photon pulses. The Stop TTL is synchronized with injection locking signal
and the Stop signal is produced by the photon pulses from the spontaneous
emission. For clarity, the plotted pulses are not strictly to scale. The
typical values: $t_{m}$ = 10 s, $T$ = 5.38 $\mu$s, and 0
$<~{}t~{}\leqslant~{}T$.

##  II Experimental system and scheme

Our experiment of ultrasensitive force detection is carried out by an
injection-locked phonon laser regarding a single trapped ion confined stably
in the harmonic potential of a surface-electrode trap (SET), see Fig. 1a. The
SET employed in our experiment was introduced previously Brownnutt2015 ;
Wan2013 ; Liu2020 ; House2008 , which is a 500-$\mu$m scale planar trap
composed of two pairs of endcap electrodes, two pairs of steering electrodes,
a pair of middle electrodes, an axial electrode and a radio-frequency
electrode. The measured secular frequencies of the SET are, respectively,
$\omega_{z}/2\pi$ = 186.02$\pm$0.01 kHz, $\omega_{x}/2\pi$ = 680.4$\pm$0.1
kHz, and $\omega_{y}/2\pi$ = 1020.3$\pm$0.5 kHz. The 40Ca+ ions, when confined
stably in the potential, stay with 800 $\mu$m above the surface of the SET.

The phonon laser is produced by the oscillation amplification of the single
40Ca+ ion stimulated by two 397 nm laser beams, one of which is a red-detuned
beam with detuning $\Delta_{1}/2\pi$ = -75 MHz and the other of which is a
blue-detuned beam with $\Delta_{2}/2\pi$ = 30 MHz. Both of the laser beams are
elaborately tuned to be with the appropriate intensity ratio $r$ =
$I_{b}/I_{r}$ = 0.5 ($I_{r}$ = 370 W/m2 and $I_{b}$ = 175 W/m2) in order to
keep the oscillation amplification stable Vahala2009 , which yields the phonon
laser with the oscillation amplitude of 17.839 $\mu$m. Besides, we have a
saturated 866 nm laser for repumping, as sketched in Fig. 1a. Since the decay
from $P_{1/2}$ to $S_{1/2}$ is much larger than that from $P_{1/2}$ to
$D_{3/2}$, the three-level system could be reasonably considered as a two-
level system Yan2019 .

In our system, we may produce the phonon laser solely regarding the $z-$axis
motional degree of freedom of the ion due to the large frequency differences
in different directions. Due to the same reason, we may lock the phonon
laser’s oscillation frequency to $\omega_{z}$, i.e., the trap frequency along
$z$ axis, by applying an appropriate injection-locking signal to AE, which has
no influence on other two directions. Generally speaking, when the injection
frequency $\omega_{i}$ is tuned to the locking range, the phonon laser’s
oscillation frequency would be fixed at $\omega_{i}$ with a very sharp
bandwidth Knunz2010 , implying that the oscillation phase $\varphi$, rather
than the oscillation amplitude $A$, is also locked. In this case, we describe
the injection force simply as $F_{0}\sin(\omega_{i}t)$, where $F_{0}$ is the
force amplitude to be detected.

The main operations of our experiment are controlled by the injection and/or
squeezing signal processing system, as sketched in Fig. 1b. The time-to-
amplitude converter (TAC) gets started from the transistor-transistor logic
(TTL) signal of the photon pulses via PMT and ended at the synchronized TTL
signal of the injection. The time resolution of the TAC $t_{r}$ is 10 ns,
which fully satisfies the requirements of the present experiment. The phonon
laser is observed from the charge-coupled-device (CCD), see the images in Fig.
1a. With the locking signal applied, we may observe the oscillation of the
phonon laser amplifying in an approximately linear way, implying that the
injection locking takes action. To acquire the values of the phase $\varphi$
and the amplitude $A$ of the oscillation, we locked the oscillation frequency
of the phonon laser, and recorded the accumulated photon counts within each
measurement time $t_{m}$, as indicated in Fig. 1c.

##  III Fitting method

For our purpose, we have developed an approach to fast acquire the value of
$A$ by fitting the recorded photons. To this end, we assume that $\varphi$ and
$A$ are constants in $t_{m}$ by ignoring the short-time noise. Under the
irradiation of the 397-nm laser beams, the scattering rate $\rho_{j}$ at time
$t$ is given by Leibfried2003 ,

$\rho_{j}(t)=\dfrac{{\Gamma}s_{j}/(4\pi)}{1+s_{j}+4\left[\frac{\Delta_{j}-k_{j}{\omega_{i}}A\cos({\omega_{i}}t+\varphi)}{\Gamma}\right]^{2}},$
(1)

where $\Gamma$ is the decay rate of $P_{1/2}$, $k_{j}$ is the wave vector,
$\Delta_{j}$ means the detuning and $s_{j}$ represents the saturation
parameter. Due to two 397-nm laser beams with different detunings in our
experiment, the total scattering rate $\rho(t)$ = $\rho_{1}(t)+\rho_{2}(t)$.
To fit the experimental photon counts, we involved the noise, such as the
projection noise, the laser power noise and etc, for which the Gaussian term
$G(t)$ is introduced in fitting the curve of $P(t)$ by a convolution function,

$P(t)=\alpha\rho(t){\ast}G(t)+\beta,$ (2)

where $G(t)$ = $(1/\sqrt{2\pi}\sigma_{t})exp[-t^{2}/(2\sigma_{t}^{2})]$ with
$\sigma_{t}$ the degree of time dispersion. $\alpha$ and $\beta$ represent the
factors regarding the measurement time $t_{m}$, the fluorescence collection
efficiency $\eta$, the background photons and the number of unit time
intervals $n$ as well as the signal-to-noise ratio (SNR) relevant to the
background light. More details about the fitting can be found in Appendix.

Figure 2: Typical fitting cases for the accumulated photon counts. (a) $A$ =
21.677 $\mu$m and $\varphi$ = 0.028 are obtained under the injection voltage
$V_{i}$ = 9.5 mV. (b) $A$ = 24.462 $\mu$m and $\varphi$ = 0.042 are acquired
under the injection voltage $V_{i}$ = 18.25 mV. The dots are experimental data
and solid curves are fitted by Eq. (2). Total photon counts $N$ =
5.353$\times$104 are recorded by 538 unit time intervals, where each
measurement time is $t_{m}$ = 10 s and we set the parameter values as
$\sigma_{t}$ = 80$t_{r}$ = 0.8 $\mu$s, $\alpha$ = 4$\times$10-5 and $\beta$ =
46.

Figure 2 presents two fitting curves with respect to the experimental data
under different injection voltages $V_{i}$. From the fitting, we acquired the
oscillation amplitude $A$ and phase $\varphi$ as constants. We observed that
tuning $\varphi$ mainly changes the position of the curve in the horizontal
axis of $t$ and varying $A$ mainly influences the height of the second peak of
the curve. This indicates that $A$ and $\varphi$ are mutually independent and
thus can be acquired simultaneously by the fitting.

##  IV Results and discussion

###  IV.1 Force sensing

The force calibration is needed prior to the force sensing due to the fact
that the injection force $F_{0}$ sensed by the ion originates from a radio-
frequency voltage $V_{i}$ on AE applied by the injection-locking signal and
$F_{0}$ should be evaluated by comparing with the counterpart regarding a DC
voltage on AE. For the latter, we have the static force $F_{c}$ =
$m\omega_{z}^{2}z$ with $z$ the position deviation of the ion due to the
applied DC voltage. In our experiment, we have measured $z$ = 12 $\mu$m when
the applied DC voltage is 3 V, implying the force $F_{c}$ = 1088.5 zN. Thus we
obtained the slope ${\partial}F_{c}/{\partial}V_{i}$ = 362.8 yN/mV.

For a harmonic oscillator, such as the trapped ion, however, the oscillation
amplitude variation due to the applied radio-frequency voltage $V_{i}$ of the
injection signal originates from the resonance absorption of the energy. But
we may reasonably consider the amplitude variation to be driven by the force
$F_{0}$ by assuming $F_{0}=F_{c}$. Since the oscillation amplitude $A$ can be
accurately measured, as discussed above, by fitting the fluorescence curve
when the oscillation frequency is locked, we obtained a linear relationship
between the injection voltage $V_{i}$ and the oscillation amplitude $A$, with
the slope ${\partial}A/{\partial}V_{i}$ = 362.1 nm/mV, see Fig. 3a. Excluding
the original oscillation amplitude without injection, i.e., 17.839 $\mu$m, we
found the relationship between the injection force and the oscillation
amplitude as ${\partial}A/{\partial}F_{0}$ = 0.9979 nm/yN. Due to frequency
drifts, the measured secular frequency along $z$ axis is $\omega_{z}/2\pi$ =
186.02$\pm$0.01 kHz, which yields the uncertainty of the measured oscillation
amplitude to be 15$\pm$2 nm regarding $V_{i}$. Therefore, the sensitivity of
our force sensor is 347$\pm$50 yN/$\sqrt{Hz}$.

###  IV.2 Squeezing

Squeezing is an effective way to reduce the noises in precision measurement
Natarajan1995 ; Burd2019 ; Majorana1997 ; Briant2003 ; Wollman2015 . In our
experiment, we try to further improve the measurement precision by employing
the approach of squeezing to suppress thermal noises regarding the oscillator,
as explored in Natarajan1995 . Compared to the quantum mechanical squeezing to
suppress quantum noises for beating standard quantum limit, the squeezing
employed here is the classical squeezing, which reduces the effect of thermal
noise by redistributing the thermal fluctuations. Since this squeezing is
produced also by doubling the motional frequency, similar to the quantum
counterpart, we may simply call it squeezing here. In our case with the
Doppler-cooled ion, this squeezing helps us improve the precision of detection
in the measurement of the smallest force.

The squeezing signal is applied on the EEs with frequency doubling and
synchronizing with respect to the injection signal. So only the motion along
$z$ axis could feel the squeezing effect. Mathematically, the uncertainty of
the oscillation can be expressed by the measured standard deviation of $A$ and
$\varphi$, i.e., $\sigma_{X}$ = $\sigma_{A}$ and $\sigma_{Y}$ =
$A_{0}\sigma_{\varphi}$, where $A_{0}$ is the oscillation amplitude in the
absence of noises. By ignoring the second-order terms in derivative as well as
the higher-order terms of frequency, we acquired the variance of $Y(t)$ as
Majorana1997 ; Briant2003 ,

$\sigma_{Y}^{2}(g,\phi)=\frac{\sigma_{Y}^{2}(0)}{1-g\cos(2\phi)},$ (3)

where $\sigma_{Y}^{2}(0)$ means the variance without squeezing and $g$ is
relevant to the trapping potential modification regarding the squeezing signal
(see Appendix for details).

We focus on the variance of the oscillation phase below since we have only
locked the oscillation phase, rather than the oscillation amplitude. Squeezing
the variance of the phase requires reducing $g\cos(2\phi)$, as indicated in
Eq. (3), which is achieved by tuning $g$ and $\phi$, respectively. Fig. 3(b,c)
present the relative variance $\sigma_{Y}^{2}(g,\phi)/\sigma_{Y}^{2}(0)$ as
functions of $g$ and $\phi$, where the red solid curves represent the
theoretical simulation using Eq. (3) and the blue dots are the experimental
results with each dot acquired by 50 measurements. From the results, we
identified the best squeezing effects occurred at 3 dB when $g\cos(2\phi)$ =
-1.

Figure 3: Detection sensitivity and smallest force measurement. (a)
Oscillation amplitude as a function of the injection voltage, where the red
dot represent the case without the injection voltage and the blue dots are
experimental data. (b) Relative variance as a function of $g$ in the case of
$\phi$ = $\pi/2$. (c) Relative variance as a function of $\phi$ in the case of
$g$ = 0.9. In (b,c) the variance without squeezing is normalized as unit, as
represented by the dashed lines. (d) Experimental results of the lower bound
of the force detection, where the blue dots are experimental data without
squeezing and the red dots with 3 dB squeezing. The horizontal axis is with
log10 scale for distinguishing the difference due to the applied squeezing.
The values of force and amplitude variation are identified from the measured
injection voltage, following the linear relationship in the text. In all the
panels, the voltage regarding the injection signal is 18.25 mV. Further
voltage of 125 mV is applied on the EEs to generate 3 dB squeezing. Each
measurement takes 10 seconds and we make 50 measurements for each data point.

In our experiment, we have observed the effect of squeezing on the oscillation
phase, rather than the oscillation amplitude. This is due to the fact that the
squeezing is relevant to the injection-locking we applied, which fixed the
phase, but not the amplitude. Compared to the frequency/phase locking, locking
amplitude is more challenging in ion-trap systems. For example, the feedback-
locking is in principle available to fix the oscillation amplitude, but hard
to achieve using current techniques since it requires higher scattering rate
and higher fluorescence collection efficiency. Nevertheless, with only
frequency/phase locked, we are able to reach the lower bound in the force
detection, as elucidated below.

###  IV.3 Lower bound of force detection

To detect the smallest force, we may reduce the injection voltage $V_{i}$
until the phonon laser is unlocked, which is judged by fitting the values
regarding the variation of the phonon laser’s oscillation amplitude with
respect to the voltage. Qualitatively, we identify the phonon laser working in
a critical point when the oscillation frequency is locked with success
probability of 90$\%$. This implies the failure of the force detection once
the locking probability is lower than 90$\%$. By this way, we have detected in
the absence of squeezing the critical voltage of 0.3 mV, corresponding to the
lower bound of the detectable force as 171.7 $\pm$ 74.3 yN, where the large
uncertainty comes from the very weak locking at the critical point and can be
resorted to the error bars, as marked in Fig. 3(d). While with 3 dB squeezing
applied, the lower bound goes further down to 86.5 $\pm$ 70.1 yN due to the
successful reach of the critical voltage of 0.15 mV, where the uncertainty is
also reduced due to squeezing. Since 3 dB is the best squeezing value in our
experiment, we consider this lower bound to be the smallest force we can
detect.

Note that, the smallest force detection in our method is determined by the
frequency locking and squeezing. Repetition of measurement just helps reducing
the detection uncertainty, but irrelevant to the lower bound. In this context,
we have measured the electric noise on the electrodes for further
understanding of the force detection lower bound. We found that the electric
noise is approximately 2 mV. This fact explains the lower bound we have
reached in our force detection, that is, the injection voltage could be sensed
by the phonon laser only when larger than 15$\%$ (8$\%$) of the total electric
noise in the absence (presence) of squeezing. This indicates that, besides
reducing the measurement variance, the squeezing also improves the signal
detection from the electric noise, favoring the sensing of the smaller force.
In this context, to further detect a smaller force, we will have to optimize
our method by suppressing the electric noises. Besides, reducing the frequency
drift would be another important effort in the future. We have found that the
frequency $\omega_{z}$ drifts for about 10 Hz during each measurement of 500
s. Since the locking range is really narrow, the frequency drift has the
strong influence on the smallest force detection which occurs at the critical
point of the frequency locking.

##  V Conclusion

In summary, we have demonstrated experimentally an ultrasensitive force sensor
of the phonon laser of a trapped ion, where injection-locking favors high
sensitivity of force detection and squeezing helps for detecting the smallest
force. Our implementation worked without the prerequisite of sideband cooling,
but efficiently suppressed thermal noises in the detection of the smallest
force. Besides, we developed a fast approach to measure the oscillation
amplitude based on counting the emitted photons. By this way, without the help
of the phase Fresnel lens as employed in Blums2018 , our force detection by
means of detecting the oscillation amplitude variation of the phonon laser is
more sensitive to the external disturbance than by detecting the ion’s
position deviation. In this context, we anticipate that our scheme could be
further optimized to achieve the force detection with better sensitivity in
the case of higher fluorescence collection efficiency, lighter ion probe and
lower electric noise. Further efforts in the future would be to suppress the
secular frequency drift resulted from the electrodes’ voltage drift and lock
the oscillation amplitude of the phonon laser. Our employed technique can also
be generalized to precisely detecting the force gradient and the surface noise
due to electric fluctuation.

## ACKNOWLEDGMENTS

This work was supported by Key Research $\&$ Development Project of Guangdong
Province under Grant No. 2020B0303300001, by National Key Research $\&$
Development Program of China under grant No. 2017YFA0304503, and by National
Natural Science Foundation of China under Grant Nos. 12074346, 12074390,
11835011, 11804375, 11804308, 91421111, 11734018.

Zhichao Liu and Yaqi Wei contributed equally to this work.

## Appendix A: Dynamics

For a harmonic oscillator with the oscillation frequency $\omega$ under
perturbation of the thermal noise, the motion is described as $z(t)$ =
$[A_{0}+\delta_{A}(t)]\sin[\omega{t}+\delta_{\varphi}(t)]$, which can be
rewritten as Natarajan1995 ,

$z(t)=X(t)\sin(\omega{t})+Y(t)\cos(\omega{t}),$ (A1)

with two orthogonal components $X(t)$ = $A_{0}+\delta_{A}(t)$ and $Y(t)$ =
$A_{0}\delta_{\varphi}(t)$. In the varying electric field, the trap secular
frequency $\omega_{z}$ in our experiment is given by

$\omega_{z}^{2}=\frac{Q}{m}\dfrac{\partial^{2}\psi(z)}{\partial{z}^{2}},$ (A2)

where $Q/m$ is the charge-to-mass ratio of 40Ca+ ion and $\psi(z)$ is the
potential along $z$ axis without squeezing.

When the classical squeezing is on, the squeezing with frequency $2\omega_{i}$
would modify the original potential, meaning
$\psi(z)[1+g_{0}\sin(2\omega_{i}t+2\phi)]$, where $g_{0}$ is the assumed gain
and $\phi$ is the relative phase with respect to the injection signal. In this
case, the trap frequency is rewritten as $\omega_{s}^{2}(t)$ =
$\omega_{z}^{2}[1+g_{0}\sin(2\omega_{i}t+2\phi)]$. For convenience of
calculation, we assume $g_{0}$ = $g\zeta/\omega_{z}$, where $g$ is
dimensionless and $g\geqslant{0}$, and $\zeta$ is the frictional damping.
Considering two 397 nm laser beams, we have the total frictional damping
$\zeta$ = $\zeta_{1}+\zeta_{2}$ with the mathematical expression as
Leibfried2003 ,

$\zeta_{j}=-\dfrac{4{\hslash}k_{j}^{2}\Delta_{j}/\Gamma}{[1+s_{j}+4(\Delta_{j}/\Gamma)^{2}]^{2}},$
(A3)

where $k_{j}$, $\Delta_{j}$ and $s_{j}$ are, respectively, the wave number,
detuning and saturation parameter of the $j$th beam of the 397-nm laser.

As the oscillation frequency of the phonon laser is locked and the squeezing
is on, we have $\omega=\omega_{i}$ and then the Langevin equation is given by
Rugar1991 ,

$\ddot{z}(t)+\zeta\dot{z}(t)+[\omega_{z}^{2}+g\zeta\omega_{z}\sin(2\omega_{i}t+2\phi)]z(t)=\dfrac{F(t)+f_{n}(t)}{m},$
(A4)

where the random force by thermal noise is
$f_{n}(t)=f_{y}(t)\sin(\omega_{i}{t})+f_{x}(t)\cos(\omega_{i}{t})$ and the
injection force is $F(t)$ = $F_{0}\sin(\omega_{i}{t})$. When the oscillation
frequency is locked, we have $\omega$ = $\omega_{i}$. Rewriting Eq. (A4) by
two orthogonal components $X(t)\sin(\omega_{i}{t})$ and
$Y(t)\cos(\omega_{i}{t})$, we obtain,

$\dot{X}(t)+\frac{\zeta}{2}\left(1+g\cos(2\phi)\right)X(t)=\frac{f_{x}(t)}{2\omega_{z}},$
(A5)
$\dot{Y}(t)+\frac{\zeta}{2}\left(1-g\cos(2\phi)\right)Y(t)=\frac{F_{0}+f_{y}(t)}{2\omega_{z}},$
(A6)

where the second-order terms and ’$\omega_{i}{t}$’ terms are neglected. Thus
the variances of $X(t)$ and $Y(t)$ related to temperature are
$\sigma_{X}^{2}(0)$ = $\sigma_{Y}^{2}(0)$ = $\frac{k_{B}T}{2m\omega_{i}^{2}}$
Majorana1997 . In our experiment, we just focus on $Y(t)$, whose variance with
the squeezing gain $g$ can be written as in Eq. 3 Briant2003 .

From Eq. (A1), we have $\sigma_{Y}$ = $A_{0}{\sigma}_{\varphi}$, where
$\sigma_{\varphi}$ is the uncertainty of the oscillation phase. Therefore, we
obtain $\sigma_{\varphi}^{2}(g,\phi)/\sigma_{\varphi}^{2}(0)$ =
$1/[1-g\cos(2\phi)]$. In the case of 0$<g\cos(2\phi)<$1, the uncertainty
increases, whereas for -1$<g\cos(2\phi)<$0, the uncertainty reduces. We have
observed experimentally that the best squeezing is 3 dB, which occurs in
$g~{}{\rightarrow}$ 1 and $\phi$ = $\pi/2$.

## Appendix B: Fluorescence collection efficiency

Improving the fluorescence collection efficiency of the force sensor is an
important technique to achieve higher sensitivity. In our experiment, the
numerical aperture of the imaging lens set is 0.36 and the transmittance is
80$\%$. The transfer efficiency of the photons by PMT is 23$\%$ and only
70$\%$ light goes through the beam splitter mirror. There are other factors in
our system for the light loss of 12$\%$. For the phonon laser, the TAC
transfer efficiency of the PMT pulses to TTL is about 66$\%$. Therefore, the
total fluorescence collection efficiency $\eta$ is about 0.25$\%$ in
theoretical prediction, which basically agrees with the measurement result
0.28$\%$. The SNR is about 2.0 and the background noise is mainly from the
diffuse reflection of the two 397-nm laser beams.

The efficiency is an important factor for sensitivity. With high efficiency,
the total measurement time $\tau$ could be shorten, where $\tau$ can be
written as the product of the single measurement time $t_{m}$ and the
measurement repetition $\varepsilon$, namely $\tau$ = $\varepsilon{t_{m}}$. In
our experiment, $\varepsilon$ = 50 and $t_{m}$ = 10 s. Therefore, the
sensitivity is given by Campbell2017 ,

$S=\frac{\delta{A}\sqrt{\tau}}{\partial{A}/\partial{F_{0}}},$ (A7)

where $\delta{A}$ is the standard deviation of the measured oscillation
amplitude. In our experiment, we have $\partial{A}/\partial{F_{0}}$ = 0.9979
nm/yN and $\delta{A}$ = 15 nm, so that the sensitivity is 347 yN/$\sqrt{Hz}$,
which could be improved by lowering the total measurement time $\tau$, i.e.,
enhancing the fluorescence collection efficiency.

## Appendix C: Drift of the secular frequency

When the oscillation frequency $\omega$ of the phonon laser is locked to the
injection frequency $\omega_{i}$, i.e., $\omega$ = $\omega_{i}$, the bandwidth
of the oscillation frequency turns to be very sharp. The locking range
$\omega_{m}$ is approximately in linear proportion to the injection force
$F_{0}$ and the central frequency of the locking range is exactly the secular
frequency $\omega_{z}$ Knunz2010 . However, there exist drifts of the voltages
applied on the electrodes, which lead to the secular frequency drifts. We have
observed experimentally the drift of the frequency for about 10 Hz per 500 s.
This drift has strong influence on the smallest force detection which occurs
at the critical point of the frequency locking. Further suppression of such
drifts would make a smaller force detection available.

## Appendix D: Fitting parameters

Before acquiring the oscillation amplitude $A$ and phase $\varphi$ by fitting
experimental data, we need to determine other parameters. The detunings of the
397-nm laser beams can be adjusted by acoustic optical modulators, and the
resonant frequency is obtained by the Lorentzian curve Li2020 ; Urabe1998 .
The densities of the 397-nm laser beams are directly measured by saturation
spectra, which are $s_{1}$ = 0.8 and $s_{2}$ = 0.4 in our experiment. The wave
vectors are approximately considered as
$k_{1}~{}\approx~{}k_{2}~{}\approx~{}k$, related to the resonant frequency
755.22 THz. The decay rate is $\Gamma/2\pi$ = 20.68 MHz.

Other three parameters $\alpha$, $\beta$ and $\sigma_{t}$ are obtained by the
fitting. We set $\sigma_{t}$ = 0.8 $\mu$s in terms of the system’s noise, and
observed the oscillation amplitude 22 $\mu$m of the phonon laser. The emitted
photons are theoretically 1.271$\times$107 within 10 seconds, while the
experimentally collected photon count $N$ is 5.353$\times$104. Since the SNR
is about 2.0, we have $\eta$ to be 0.28$\%$. Then we acquired the values of
$\alpha$ and $\beta$ by $\alpha$ = $\eta{t_{m}}/n$ and $\beta$ =
$N/[n(1+SNR)]$, which led to the values of $A$ and $\varphi$.

## References

  * (1) O. Hosten, N. J. Engelsen, R. Krishnakumar, M. A. Kasevich, Measurement noise 100 times lower than the quantum-projection limit using entangled atoms. Nature 529, 505-508 (2016).
  * (2) S. Schreppler, N. Spethmann, N. Brahms, T. Botter, M. Barrios, D. M. Stamper-Kurn, Optically measuring force near the standard quantum limit. Science 344, 1486-1489 (2019).
  * (3) H. J. Mamin, D. Rugar, Sub-attonewton force detection at millikelvin temperatures. Appl. Phys. Lett. 79, 3358 (2001).
  * (4) F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, J. D. Teufel, Quantum nondemolition measurement of a nonclassical state of a massive object. Phys. Rev. X 5, 041037 (2001).
  * (5) C. M. Caves, Quantum-mechanical noise in an interferometer. Phys. Rev. D 23, 8 (1980).
  * (6) C. L. Degen, F. Reinhard. P. Cappellaro, Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).
  * (7) S. Kotler, N. Akerman, Y. Glickman, A. Keselman, R. Ozeri, Single-ion quantum lock-in amplifier. Nature 473, 61-65 (2011).
  * (8) E. Kim, A. Safavi-Naini, D. A. Hite, K. S. McKay, D. P. Pappas, P. F. Weck, and H. R. Sadeghpour. Electric-field noise from carbon-adatom diffusion on a Au(110) surface: First-principles calculations and experiments. Phys. Rev. A 95, 033407 (2017).
  * (9) K. A. Gilmore, J. G. Bohnet, B. C. Sawyer, J.W. Britton, and J. J. Bollinger, Amplitude sensing below the zero-point fluctuations with a two-dimensional trapped-ion mechanical oscillator. Phys. Rev. Lett. 118, 263602 (2017).
  * (10) M. Affolter, K. A. Gilmore, J. E. Jordan and J. J. Bollinger. Phase-coherent sensing of the center-of-mass motion of trapped-ion crystals. Phys. Rev. A 102, 052609 (2020).
  * (11) K. A. Gilmore, M. Affolter, R. J. Lewis-Swan, D. Barberena, E. Jordan, A. M. Rey and J. J. Bollinger. Quantum-enhanced sensing of displacements and electric fields with large trapped-ion crystals. arXiv:2103.08690.
  * (12) S. Knünz, M. Herrmann, V. Batteiger, G. Saathoff, T. W. Hänsch, K. Vahala, T. Udem, Injection-lock of a trapped-ion phonon laser. Phys. Rev. Lett. 105, 013004 (2010).
  * (13) M. J. Biercuk, H. Uys, J. W. Britton, A. P. VanDevender, J. J. Bollinger, Ultrasensitive detection of force and displacement using trapped ions. Nat. Nanotechnol. 5, 646-650 (2010).
  * (14) R. Shaniv, R. Ozeri, Quantum lock-in force sensing using optical clock Doppler velocimetry. Nat. Commun 8, 14157 (2017).
  * (15) V. Blūms, M. Piotrowski, M. I. Hussain, B. G. Norton, S. C. Connell, S. Gensemer, M. Lobino, E. W. Streed, A single-atom 3D sub-attonewton force sensor. Sci. Adv. 4, 4453 (2018).
  * (16) G. Araneda, S. Walser, Y. Colombe, D. B. Higginbottom, J. Volz, R. Blatt, A. Rauschenbeute, Wavelength-scale errors in optical localization due to spin-orbit coupling of light. Nat. Phys. 15, 17-21 (2019).
  * (17) V. Natarajan, F. DiFilippo, D. E. Pritchard, Classical squeezing of an oscillator for subthermal noise operation. Phys. Rev. Lett. 74, 2855 (1995).
  * (18) K. Vahala, M. Herrmann, S. Knünz, V. Batteiger, G. Saathoff, T. W. Hänsch and T. Udem, A phonon laser. Nat. Phys. 5, 682-686 (2009).
  * (19) I. S. Grudinin, H. Lee, O. Painter, K. J. Vahala, Phonon laser action in a tunable two-level system. Phys. Rev. Lett. 104, 083901 (2010).
  * (20) M. Ip, A. Ransford, A. M. Jayich, X. Long, C. Roman, and W. C. Campbell, Phonon lasing from optical frequency comb illumination of trapped ions. Phys. Rev. Lett 121, 043201 (2018).
  * (21) B. He, L. Yang, M. Xiao, Dynamical phonon laser in coupled active-passive microresonators. Phys. Rev. A 94, 031802 (2016).
  * (22) M. R. Hush, W. Li, S. Genway, I. Lesanovsky, A. D. Armour, Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers. Phys. Rev. A 91, 061401 (2015).
  * (23) F. Dominguez, I. Arrazola, J. Doménech1, J. S. Pedernales, L. Lamata, E. Solano, D. Rodriguez1, A single-ion reservoir as a high-sensitive sensor of electric siqnals. Sci. Rep. 7, 8336 (2017).
  * (24) M. Brownnutt, M. Kumph, P. Rabl, R. Blatt, Ion-trap measurements of electric-field noise near surfaces. Rev. Mod. Phys. 87, 1419 (2015).
  * (25) W. Wan, H. Y. Wu, L. Chen, F. Zhou, S. J. Gong, and M. Feng, Demonstration of motion transduction in a single-ion nonlinear mechanical oscillator. Phys. Rev. A 89, 063401 (2013).
  * (26) Z. C. Liu, L. Chen, Ji Li, H. Zhang, C. Li, F. Zhou, S. L. Su, L. L. Yan, M. Feng, Structural phase transition of the ion crystals embedded in an optical lattice. Phys. Rev. A 102, 033116 (2020).
  * (27) M. G. House, Analytic model for electrostatic fields in surface-electrode ion traps. Phys. Rev. A 78, 033402 (2008).
  * (28) L. L. Yan, S. L. Su, and M. Feng, Analytical investigation of one-dimensional Doppler cooling of trapped ions with $\Lambda$-type configuration. Phys. Rev. A 100, 033418 (2019).
  * (29) D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281 (2003).
  * (30) E. Majorana, Y. Ogawa, Mechanical thermal noise in coupled oscillators. Phys. Lett. A 233, 162-168 (1997).
  * (31) T. Briant, P. F. Cohadon, M. Pinard, A. Heidmanna, Optical phase-space reconstruction of mirror motion at the attometer level. Eur. Phys. J. D 22, 131-140 (2003).
  * (32) E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, K. C. Schwab, Quantum squeezing of motion in a mechanical resonator. Phys. Rev. A 349, 952-955 (2015).
  * (33) S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, D. T. C. Allcock, Quantum amplification of mechanical oscillator motion. Science 364, 1163-1165 (2019).
  * (34) J. Li, L. Chen, Y. H. Chen, Z. C. Liu, H. Zhang. and M. Feng, Three-Dimensional Compensation for Minimizing Heating of the Ion in Surface-Electrode Trap. Chin. Phys. Lett. 37, 053701 (2020).
  * (35) S. Urabe, M. Watanabe, H. Imajo, K. Hayasaka, U. Tanaka and R. Ohmukai, Observation of Doppler sidebands of a laser-cooled Ca+ ion by using a low-temperature-operated Laser diode. Apply. Phys. B 67, 223 (1998).
  * (36) D. Rugar and P. Griitter, Mechanical parametric amplification and thermomechanical noise squeezing. Phys. Rev. Lett. 67, 6 (1991).
  * (37) W. C. Campbell, and P. Hamilton, Rotation sensing with trapped ions. J. Phys. B 50, 064002 (2017).


# Distant entanglement via photon hopping in a coupled magnomechanical system

Amjad Sohail  Department of Physics, Government College University, Allama
Iqbal Road, Faisalabad 38000, Pakistan  Electrical and Computer Engineering
Department, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates  Jia-
Xin Peng  State Key Laboratory of Precision Spectroscopy, Quantum Institute
for Light and Atoms, Department of Physics, East China Normal University,
Shanghai 200062, China  Abdelkader Hidki  LPTHE, Department of Physics,
Faculty of Sciences, Ibn Zohr University, Agadir, Morocco  S. K. Singh
Graphene and Advanced 2D Materials Research Group (GAMRG), School of
Engineering and Technology, Sunway University, No. 5, Jalan Universiti, Bandar
Sunway, 47500 Petaling Jaya, Selangor, Malaysia

###### Abstract

We theoretically propose a scheme to generate distant bipartite entanglement
between various subsystems in coupled magnomechanical systems where both the
microwave cavities are coupled through single photon hopping parameter. Each
cavity also contains a magnon mode and phonon mode and this gives five
excitation modes in our model Hamiltonian which are cavity-1 photons, cavity-2
photons, magnon, and phonon modes in both YIG spheres. We found that
significant bipartite entanglement exists between indirectly coupled
subsystems in coupled microwave cavities for an appropriate set of parameters
regime. Moreover, we also obtain suitable cavity and magnon detuning
parameters for a significant distant bipartite entanglement in different
bipartitions. In addition, it can be seen that a single photon hopping
parameter significantly affects both the degree as well as the transfer of
quantum entanglement between various bipartitions. Hence, our present study
related to coupled microwave cavity magnomechanical configuration will open
new perspectives in coherent control of various quantum correlations including
quantum state transfer among macroscopic quantum systems

##  I Introduction

Quantum entanglement is one of the most fascinating phenomena in quantum
mechanics which is also unique property of quantum manybody systems [1, 2]. In
the beginning era of quantum technology, seminal theoretical and experimental
investigations mainly explored only microscopic systems, such as atoms, ions,
etc to obtain quantum entanglement [3]. However, there is no clear physical
law that states that quantum entanglement can only occur in microscopic
systems. In 2007, Vitali et al. first proposed the entanglement between a
single cavity mode and a vibrating mirror, which is the beginning of
macroscopic entanglement research [4]. Subsequently, the study of macroscopic
quantum entanglement phenomena based on optical mechanical systems received
widespread attention, including the entanglement of two vibrating mirrors [5,
6, 7, 8, 9], entanglement of multiple cavity modes coupled to vibrating
objects [10, 11, 12], entanglement in Laguerre-Gaussian cavity system [13, 14,
15, 16, 17].

Recently, ferrimagnetic materials have provided a powerful platform for
studying the essence of magnetic systems [18, 19, 20, 21]. Yttrium iron garnet
(YIG) crystal is one of the most representative materials in low damping
magnetic materials owing to its extremely high spin density and excellent
integration performance [22, 23]. Particularly, the Kittel mode in the YIG
sphere and the microwave cavity photons can be coupled to achieve the vacuum
Rabi splitting and cavity-magnon polaritons [24, 25, 26]. This induced the
birth of magnon cavity QED, which provides a promising platform for the study
of strong interactions between light and matter. Naturally, many interesting
quantum phenomena have been studied based on cavity magnetic systems, such as
magnoninduced transparency [27, 28, 29, 30], coherent feedback [31], magnon
dark modes [32], bistability [33, 34], the magnon Kerr effect [35, 36, 37, 38,
39], microwave-optical conversion [40], magnon blocking [41, 42], and so on.

It is worth mentioning that, Li Jie first studied magnon-photon-phonon
entanglement based on cavity magnetic system in 2018, which opens a new
subfield within the field of quantum entanglement [26]. Subsequently, them
presentd a scheme to entangle two microwave fields by using the nonlinear
magnetostrictive interaction [43, 44]. In addition, the macroscopic
entanglement between two YIG spheres has also been studied [45]. Considering
that the the magnon Kerr effect may be boost the quantum effect, this
stimulates the research on enhancing the entanglement between two magnon modes
by using the Kerr effect [46, 47]. Moreover, the photon-magnon entanglement is
improved by using parametric amplifier [48, 49] and squeezing effect [50, 51].
More interestingly, the remote magnon entanglement between two massive
ferrimagnetic spheres [52, 53] and robust optical entanglement are also
implemented in cavity optomagnonics system [54]. Besides the entanglement
between magnon and ordinary cavity mode, the entanglement between magnon and
Laguerre-Gaussian cavity mode is also been studied theoretically [17].
Furthermore, distant entanglement via photon hoping between different modes
has been of great interest for storing/sharing quantum information. Recently,
Chen et. al. studied the perfect transferring of enatnaglement and quntum
steering between different modes in coupled cavity magnomechanical system
[55]. In addition, Dilawaiz et. al. investigate the entanglement between a YIG
sphere and an atomic ensamble via photon hoping in coupled microwave cavities
by [56]. Therefore, researchers pay more attention to investigate the quantum
corelation via photon hoping among different/distant bipartitions, Motivated
by these developments, we consider coupled magnomechanical system to
investigate weather we can generate distant enatnglement between different bi-
partitions. Therefore, we emphsis on the underlying physical understanding of
the generation of distant entanglements via photon hoping. Furthermore, such a
well-designed coupled magnomechanical system can be utilized to create and
transfer continuous variable entanglement between different distant bosonic
modes.

##  II The Model

The magnomechanical system under consideration consists of two MW cavities
connected through single photon hoping factor $\Gamma$. Each cavity contain a
magnon mode $m$ and a phonon mode $b$ as shown in Fig. 1. The magnons are
considered to be quasiparticles which are incorporated by a collective
excitation of a large number of spins inside a ferrimagnet, e.g., a YIG sphere
[57]. The magnetic dipole interaction enables the coupling between the magnon
and the MW field. The orientation of YIG sphere inside each cavity field is in
the region of the maximum magnetic field (See Fig. 1). At the YIG sphere site,
the magnetic field of the cavity mode is along the x axis while the drive
magnetic field is along the y direction). Furthermore, the bias magnetic field
is set in the z direction. In addition, the magnon and phonon modes are
coupled to each other via magnetostrictive force, which yields the magnon-
phonon coupling [58, 59]. The magnetostrictive interaction depends on the
resonance frequencies of the magnon and phonon modes [60]. In the current
study, we assumed the frequency magnon to be much larger than mechanical
frequency, which helps to set up the strong dispersive magnon-phonon
interaction [57, 61]. The Hamiltonian of the magnomechanical system can be
written as

$\displaystyle H/\hbar$ $\displaystyle=$ $\displaystyle H_{0}+H_{int}+H_{d},$
(1)

where

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2}\left[\omega_{k}c_{k}c_{k}^{{\dagger}}+\omega_{m_{k}}m_{k}^{{\dagger}}m_{k}+\frac{\omega_{b_{k}}}{2}\left(q_{k}^{2}+p_{k}^{2}\right)\right],$
(2) $\displaystyle H_{int}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2}\left[[g_{mb}m^{{\dagger}}m_{k}q_{k}+g_{k}\left(c_{k}m^{{\dagger}}+c_{k}^{{\dagger}}m\right)\right]+\Gamma(c_{1}c_{2}^{{\dagger}}+c_{1}^{{\dagger}}c_{2}),$
(3) $\displaystyle H_{d}$ $\displaystyle=$ $\displaystyle
i\Omega\sum_{k=1}^{2}\left[]m_{k}^{{\dagger}}e^{-i\omega_{0}t}-m_{k}e^{i\omega_{0}t}\right],$
(4)

where $c_{k}\left(c_{k}^{{\dagger}}\right)$ and
$m_{k}\left(m_{k}^{{\dagger}}\right)$ are the annihilation (creation) operator
of the the $k$ cavity and magnon mode, respectively. Furthermore, $q_{k}$ and
$p_{k}$ are the position and momentum quadratures of the respective mechanical
mode of the magnon. In addition $\omega_{k}$, $\omega_{b}$ and $\omega_{m}$
are the resonance frequencies of the cavity mode $k$, mechanical mode and the
magnon mode. The magnon frequency $\omega_{m}$ can be flexibly adjusted by the
bias magnetic field $B$ via $\omega_{m}=\gamma_{0}B$. Here $\gamma_{0}$ is the
gyromagnetic ratio. The optomagnonical coupling is theoretically given by

$\displaystyle\Gamma_{k}$ $\displaystyle=$
$\displaystyle\mathcal{V}\frac{c}{n_{r}}\sqrt{\frac{2}{\rho_{spin}V_{YS}}},$
(5)

where $\mathcal{V}$, $n_{r}$, $\rho_{spin}$ and $V_{YS}=\frac{4\pi r^{3}}{3}$
are, respectively, the YIG sphere’s Verdet constant, the refractive index, the
spin density and the volume of the YIG sphere [62]. We considered strong
coupling regime i.e., the coupling between the cavity mode $k$ with magnon
mode $\Gamma_{k}$ can be larger than the decay rate of the magnon and the
cavity modes, $\Gamma_{k}>\kappa_{k}$, $\kappa_{m}$ [63, 64, 65, 66]. Here,
$g_{mb}$ denotes single-magnon magnomechanical coupling rate which is
considered to be very small but can be enhanced by directly driving the YIG
sphere with a MW source. The Rabi frequency
$\Omega=\left(\sqrt{5}/4\right)\gamma_{0}\sqrt{N_{spin}}B_{0}$ [67, 68]
represents the coupling strength of the drive field with frequency
$\omega_{0}$), amplitude $B_{0}=3.9\times 10^{-9}$T, where
$\gamma_{0}=28$GHz/T and the total number of spins $N_{spin}=\rho V_{YS}$ with
the spin density of the YIG $\rho_{spin}=4.22\times 10^{27}m^{-3}$. In
addition, it is noteworthy to mention here that collective motion of the spins
are truncated to form bosonic operators $m$ and $m^{\dagger}$ via the
Holstein-Primakoff transformation and further, the Rabi frequency $\Omega$ is
derived under the basic assumption of the low-lying excitations $2Ns\gg\langle
m^{\dagger}m\rangle$, where $s=\frac{5}{2}$ is the spin number of the ground
state Fe3+ ion in YIG.

Figure 1: (Color Online) (a) Schematic diagram of the coupled cavity
magnomechanical system in which each cavity mode contain a magnon mode in a
YIG sphere couples that interact with the microwave cavity modes via magnetic
dipole interaction and with phonon mode via magnetostrictive interaction. The
magnetic field of the each microwave cavity modes is set to be along the
x-direction, while the drive magnetic field (bias magnetic field) is
considered along y-direction (z-direction). (b) The linear coupling diagram of
each cavity magnomechanical system is shown. The two cavity modes are coupled
via photon hoping $\Gamma$, while a cavity mode photon $c_{1}$ ($c_{2}$) is
coupled to the magnon mode $m_{1}$ ($m_{2}$), with coupling strength $g_{1}$
($g_{2}$), which then coupled to a phonon mode $b_{1}$ ($b_{2}$) to with
magnomechanical coupling strength $g_{m1}$ ($g_{m2}$).

In the rotating wave approximation at the drive frequency $\omega_{0}$, he
Hamiltonian of the system can be written as

$\displaystyle H/\hbar$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2}\left[\Delta_{k}c_{k}c_{k}^{{\dagger}}+\Delta_{m_{k}}m_{k}^{{\dagger}}m_{k}+\frac{\omega_{b_{k}}}{2}\left(q_{k}^{2}+p_{k}^{2}\right)+g_{mk}m_{k}^{{\dagger}}m_{k}q_{k}+g_{k}\left(c_{k}m_{k}^{{\dagger}}+c_{k}^{{\dagger}}m_{k}\right)+i\Omega\left(m_{k}^{{\dagger}}-m_{k}\right)\right]$
$\displaystyle+\Gamma(c_{1}c_{2}^{{\dagger}}+c_{1}^{{\dagger}}c_{2}),$

where $\Delta_{k}=\omega_{k}-\omega_{0k}$ ($k=1,2$) and
$\Delta_{m_{k}}=\omega_{m}-\omega_{0k}$.

##  III Quantum dynamics and entanglement of the magnomechanical system

We now start to obtain the equations for the dynamics of this magnomechanical
system. By incorporating the effect of noises and dissipations, the following
set of quantum Langevin equations for the magnomechanical system can be
obtained:

$\displaystyle\dot{q}_{k}$ $\displaystyle=$
$\displaystyle\omega_{b_{k}}p_{k},$ (7) $\displaystyle\dot{p}_{k}$
$\displaystyle=$
$\displaystyle-\omega_{b}q_{k}-\gamma_{b}p_{k}-g_{mk}m^{{\dagger}}_{k}m_{k}+\xi_{k},$
(8) $\displaystyle\dot{c}_{k}$ $\displaystyle=$
$\displaystyle-\left(i\Delta_{k}+\kappa_{k}\right)c_{k}-ig_{k}m_{k}+\Gamma
c_{j}+\sqrt{2\kappa_{a}}c_{k}^{in},(j\neq k)$ (9) $\displaystyle\dot{m}_{k}$
$\displaystyle=$
$\displaystyle-\left(i\Delta_{m_{k}}+\kappa_{m}\right)m_{k}-ig_{k}c_{k}-ig_{mk}m_{k}q_{k}+\Omega_{k}+\sqrt{2\kappa_{m}}m^{in},$
(10)

where $k_{k}$($k_{m}$) is the decay rate of the $k^{th}$ cavity mode (magnon
mode) while $\gamma_{b}$ denotes the mechanical damping rate. $\xi$, $m^{in}$
and $c_{k}^{in}$ are input noise operators for the mechanical, magnon and
cavity modes respectively. These noise operators are characterized by the
following correlation functions [69]:

$\displaystyle\left\langle\xi(t)\xi(t^{\prime})\right\rangle+\left\langle\xi(t^{\prime})\xi(t)\right\rangle/2$
$\displaystyle=$
$\displaystyle\gamma_{b}[2n_{b}(\omega_{b})+1]\delta(t-t^{\prime}),$ (11)
$\displaystyle\left\langle
c_{k}^{in}(t)c_{k}^{in{\dagger}}(t^{\prime})\right\rangle$ $\displaystyle=$
$\displaystyle[n_{k}(\omega_{k})+1]\delta(t-t^{\prime}),$ (12)
$\displaystyle\left\langle
c_{k}^{in{\dagger}}(t)c_{k}^{in}(t^{\prime})\right\rangle$ $\displaystyle=$
$\displaystyle n_{k}(\omega_{k})\delta(t-t^{\prime}),$ (13)
$\displaystyle\left\langle m^{in{\dagger}}(t)m^{in}(t^{\prime})\right\rangle$
$\displaystyle=$ $\displaystyle n_{m}(\omega_{m})\delta(t-t^{\prime}),$ (14)
$\displaystyle\left\langle m^{in}(t)m^{in{\dagger}}(t^{\prime})\right\rangle$
$\displaystyle=$ $\displaystyle[n_{m}(\omega_{m})+1]\delta(t-t^{\prime}),$
(15)

The equilibrium mean thermal photon, magnon, and phonon numbers are
$n_{f}(\omega_{f})=[\exp(\frac{\hbar\omega_{f}}{k_{b}T})-1]^{-1}$
$[f=k(k=1,2),m,b]$, where $T$ is the environmental temperature and $k_{b}$ the
Boltzmann constant.

If the magnon mode is strongly driven, then we must have $\left|\left\langle
m\right\rangle\right|\gg 1$. In addition, the two MW cavity fields show large
amplitudes due to the cavity-magnon beam splitter interactions. This permits
us to linearize the above quantum Langevin equations by writing any operator
as a sum of average value plus its fluctuation i.e., $o=\left\langle
o\right\rangle+\delta o$, $(o=p,q,c_{k},m)$ and substitute it into Eq.(7-10).
The average values of the dynamical operators are obtained as

$\displaystyle\left\langle p_{k}\right\rangle$ $\displaystyle=$ $\displaystyle
0,$ (16) $\displaystyle\left\langle q_{k}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{-g_{mk}}{\omega_{b}}\left|\left\langle
m_{k}\right\rangle\right|^{2},$ (17) $\displaystyle\left\langle
m_{j}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\Omega_{j}-ig_{k}\left\langle
c_{k}\right\rangle}{i\Delta_{m_{k}}+\kappa_{m_{k}}},$ (18)
$\displaystyle\left\langle c_{1}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{ig_{1}\Omega_{1}\alpha_{2}-\Gamma
g_{2}\Omega_{2}(\kappa_{m_{1}}+i\Delta_{m_{1}})}{\alpha_{1}\alpha_{2}+\Gamma^{2}\left(i\Delta_{m_{1}}+\kappa_{m_{1}}\right)\left(i\Delta_{m_{2}}+\kappa_{m_{2}}\right)},$
(19) $\displaystyle\left\langle c_{2}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{ig_{2}\Omega_{2}\alpha_{1}-\Gamma
g_{1}\Omega_{1}(\kappa_{m_{2}}+i\Delta_{m_{2}})}{\alpha_{1}\alpha_{2}+\Gamma^{2}\left(i\Delta_{m_{2}}+\kappa_{m_{2}}\right)\left(i\Delta_{m_{1}}+\kappa_{m_{1}}\right)},$
(20)

where
$\alpha_{j}=(i\Delta_{i}+\kappa_{i})(i\Delta_{m_{i}}+\kappa_{m_{i}})+g_{i}^{2}$
, $\Delta_{m}=\Delta_{m_{0}}+g_{mb}\left\langle q\right\rangle$ is the
effective magnon mode detuning which includes the slight shift of frequency
due to the magnomechanical interaction.

Now, we introduce the quadrature for the linearised quantum Langevin equations
describing fluctuations are: $\delta x=\frac{1}{\sqrt{2}}(\delta m-\delta
m^{{\dagger}})$, $\delta y=\frac{1}{\sqrt{2}i}(\delta m-\delta
m^{{\dagger}})$, $\delta X_{k}=\frac{1}{\sqrt{2}}(\delta c_{k}-\delta
c_{k}^{{\dagger}})$, $\delta Y_{k}=\frac{1}{\sqrt{2}i}(\delta c_{k}-\delta
c_{k}^{{\dagger}})$ can be written in concise form as

$\dot{\mathcal{F}}(t)=\mathcal{M}\mathcal{F}(t)+\mathcal{N}(t),$ (21)

where $\mathcal{F}(t)$ and $\mathcal{N}(t)$ are, respectively, the quantum the
fluctuation and input noise vectors and are given by:

$\displaystyle\mathcal{F}(t)$ $\displaystyle=$ $\displaystyle[\delta
C_{XY}(t),\delta M_{xy}(t),\delta Q_{qp}(t)]^{T},$
$\displaystyle\mathcal{N}(t)$ $\displaystyle=$
$\displaystyle[\mathcal{N}_{XY},\mathcal{N}_{xy},\mathcal{N}_{qp}]$

where

$\displaystyle\delta C_{XY}(t)$ $\displaystyle=$ $\displaystyle\delta
X_{1}(t),\delta Y_{1}(t),\delta X_{2}(t),\delta Y_{2}(t)$ $\displaystyle\delta
M_{xy}(t)$ $\displaystyle=$ $\displaystyle\delta x_{1}(t),\delta
y_{1}(t),\delta x_{2}(t),\delta y_{2}(t)$ $\displaystyle\delta Q_{qp}(t)$
$\displaystyle=$ $\displaystyle\delta q_{1}(t),\delta p_{1}(t),\delta
q_{2}(t),\delta p_{2}(t)$ $\displaystyle\mathcal{N}_{XY}$ $\displaystyle=$
$\displaystyle\sqrt{2k_{1}}X_{1}^{in}(t),\sqrt{2k_{1}}Y_{1}^{in}(t),\sqrt{2k_{2}}X_{2}^{in}(t),\sqrt{2k_{2}}Y_{2}^{in}(t)$
$\displaystyle\mathcal{N}_{xy}$ $\displaystyle=$
$\displaystyle\sqrt{2k_{m}}x_{1}^{in}(t),\sqrt{2k_{m}}y_{1}^{in}(t),\sqrt{2k_{m}}x_{2}^{in}(t),\sqrt{2k_{m}}y_{2}^{in}(t)$
$\displaystyle\mathcal{N}_{qp}$ $\displaystyle=$ $\displaystyle
0,\xi_{1}(t),0,\xi_{2}(t)$

Furthermore, the drift matrix $\mathcal{M}$ can be written as

$\mathcal{M}=\left[\begin{array}[]{cccccccccccc}-\kappa_{1}&\Delta_{1}&0&\Gamma&0&g_{1}&0&0&0&0&0&0\\\
-\Delta_{1}&-\kappa_{1}&-\Gamma&0&-g_{1}&0&0&0&0&0&0&0\\\
0&\Gamma&-\kappa_{2}&\Delta_{2}&0&0&0&g_{2}&0&0&0&0\\\
-\Gamma&0&-\Delta_{2}&-\kappa_{2}&0&0&-g_{2}&0&0&0&0&0\\\
0&g_{1}&0&0&-\kappa_{m_{1}}&\Delta_{m_{1}}&0&0&-G_{1}&0&0&0\\\
-g_{1}&0&0&0&-\Delta_{m_{1}}&-\kappa_{m_{1}}&0&0&0&0&0&0\\\
0&0&0&g_{2}&0&0&-\kappa_{m_{2}}&\Delta_{m_{2}}&0&0&-G_{2}&0\\\
0&0&-g_{2}&0&0&0&-\Delta_{m_{2}}&-\kappa_{b_{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&\omega_{b_{1}}&0&0\\\
0&0&0&0&0&G_{1}&0&0&-\omega_{b_{1}}&-\gamma_{b_{1}}&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&\omega_{b_{2}}\\\
0&0&0&0&0&0&0&0&G_{2}&0&0&\gamma_{b_{2}}\end{array}\right],$ (22)

where $G_{mb}=i\sqrt{2}g_{mb}\left\langle m\right\rangle$ is the effective
magnomechanical coupling rate. By using Eq. (4), one can notice that effective
magnomechanical coupling rate can be increased by applying strong magnon
drive.

Next, we discuss the quantum correlation of bipartite subsystems with a
special emphasis on the entanglement of two indirectly coupled modes and the
two MW fields in the steady-state. The stability of the proposed system is the
first prerequisite for the effectiveness of our scheme. According to Routh-
Hurwitz criterion [70], the system is stable only if the real part of the all
the eigenvalues of the drift matrix $\mathcal{M}$ are negative. Hence, we
start our analysis by determining eigenvalues of the drift matrix
$\mathcal{M}$ (i.e., $|\mathcal{M}-\lambda_{\mathcal{M}}\mathbb{1}|=0$) and
make sure the stability condition are all satisfied in the following section
(see Appendix A). The magnomechanical system presented here is characterized
by $8\times 8$ covariance matrix V with its entries

$V_{ij}(t)=\frac{1}{2}\left\langle
F_{i}(t)F_{j}(t^{\prime})+F_{j}(t^{\prime})F_{i}(t)\right\rangle,$ (23)

The covariance matrix of the magnomechanical system can be obtained from the
steady state Lyapunov equation [71, 72]

$\mathcal{M}V+V\mathcal{M}^{T}=-\mathcal{D},$ (24)

where $\mathcal{D}=$
diag$[0,\gamma_{b}\left(2n_{b}+1\right),\kappa_{m}\left(2n_{m}+1\right),\kappa_{m}(2n_{m}+1)$
$\kappa_{1}\left(2n_{1}+1\right)$ ,$\kappa_{1}\left(2n_{1}+1\right)$,
$\kappa_{2}\left(2n_{2}+1\right),\kappa_{2}\left(2n_{2}+1\right)]$, is a
diagonal matrix which is called diffusion matrix and characterizes the noise
correlations. The Lyapunov Eq. (24) as a linear equation for $V$ can be easily
solved. Using the Simon condition for Gaussian states, we calculate the
entanglement of the steady state [73, 74, 75, 76, 72].

$E_{N}=\max[0,-\ln 2\eta^{-}],$ (25)

where $\eta^{-}=$min
eig$|\bigoplus^{2}_{j=1}(-\sigma_{y})\widetilde{\mathcal{V}_{4}}|$ is the
minimum symplectic eigenvalue of covariance matrix and is
$\widetilde{\mathcal{V}_{4}}=\varrho_{1|2}\mathcal{V}_{in}\varrho_{1|2}$,
where $\mathcal{V}_{in}$ is a $4\times 4$ matrix of any two subsystems which
can easily be obtain by neglecting the uninteresting rows and columns in
$\mathcal{V}_{4}$. $\varrho_{1|2}=\sigma_{z}\bigoplus 1$=diag$(1,-1,1,1)$ is
the matrix which characterizes the partial transposition at the level of
covariance matrices. Here, $\sigma_{y}$ and $\sigma_{z}$ are the pauli spin
matrices. Furthermore, a nonzero logarithmic negativity i.e., $E_{N}>0$
defines the presence of bipartite entanglement in our cavity magnomechanical
system.

##  IV Results and Discussion

Parameters | Symbol | Value | Parameters | Symbol | Value
---|---|---|---|---|---
Phonon frequency | $\omega_{b}$ |  $2\pi\times$10 MHz | Cavity frequency | $\omega_{1}=\omega_{2}=\omega_{a}$ |  $2\pi\times$ 10 GHz
Cavity decay rates | $\kappa_{1}=\kappa_{2}=\kappa$ |  $2\pi\times$ 1 MHz | Magnon decay rate | $\kappa_{m}$ |  $2\pi\times$1 MHz
Mechanical damping rate | $\gamma_{b}$ |  $2\pi\times$100 Hz | Magnon-Microwave couplings | $\Gamma_{1}=\Gamma_{2}=\Gamma$ |  $2\pi\times$3.2 MHz
Magnomechanical coupling | $g_{mb}$ |  $2\pi\times$0.3 Hz | Drive Magnetic Field | B |  $3.9\times 10^{-5}$T
YIG Sphere Diameter | D | 250 $\mu$m | Temperature | T | 10 mK
Power | $\wp=\frac{B^{2}\pi r^{2}c}{2\mu^{2}}$ | 9.8 mW | Spin density | $\rho$ | $4.22\times 10^{27}m^{-3}$
Table 1: Parameters used in recent experiments for the mechanical resonators Bipartite Subsystem | Entanglement Symbol | Bipartite Subsystem | Entanglement Symbol
---|---|---|---
Cavity 1-Cavity 2 | $E_{c-c}^{N}$ |  |
Cavity 1-magnon 2 | $E^{N}_{c_{1}-m_{2}}$ | Cavity 2-magnon 1 | $E^{N}_{c_{2}-m_{1}}$
Cavity 1-phonon 2 | $E^{N}_{c_{1}-b_{2}}$ | Cavity 2-phonon 1 | $E^{N}_{c_{2}-b_{1}}$
Table 2: Adopted notation for the different bipartite subsystem entanglement.
Figure 2: (Color Online) Density plot of bipartite entanglement in (a),(d)
$E_{c_{1}-c_{2}}^{N}$; in (b),(e) $E_{c_{1}-m_{2}}^{N}=E_{c_{2}-m_{1}}^{N}$
and in (c),(f) $E_{c_{1}-b_{2}}^{N}=E_{c_{2}-b_{1}}^{N}$ versus cavity
detunings $\Delta_{1}/\omega_{b}$ and $\Delta_{2}/\omega_{b}$ in (a)-(c) for
$\Delta_{m_{1}}=\Delta_{m_{2}}=\omega_{b}$ whereas varying both magnon
detunings $\Delta_{m_{1}}/\omega_{b}$ and $\Delta_{m_{2}}/\omega_{b}$ in
(d)-(f). We use optimum values of of $\Delta_{1}$ and $\Delta_{2}$ for
(d)-(f). The other parameters are given in Table 1.

In this section we are going to discuss in details the results of bipartite
entanglements as we have six different modes in this coupled cavity
Magnomechnical system. So, we can get bipartite entanglement in any of two
modes however the most significant part of our study is to investigate the
bipartite entanglement present in various spatially distant subsystems which
we have summarised in Table 2 with symbols.
In Fig. 2, we present five different distant bipartite entanglements as a
function of dimensionless cavity detuning for first cavity
$\Delta_{1}/\omega_{b}$ and second cavity $\Delta_{2}/\omega_{b}$. When both
the magnon detuning is kept in resonant with blue sideband regime i.e.
$\Delta_{m_{1}}=\Delta_{m_{2}}=\omega_{b}$ it can be seen that bipartite
entanglement between two cavity modes $E_{c_{1}-c_{2}}^{N}$ become maximum for
$\Delta_{1}=\Delta_{2}=-0.5\omega_{b}$ although even if both the cavities are
resonant with blue sideband regime i.e. $\Delta_{1}=\Delta_{2}=\omega_{b}$ we
have significant amount of bipartite entanglement in $E_{c_{1}-c_{2}}^{N}$ as
shown in Fig. 2(a). In Fig.2(b) we study the bipartite entanglement
$E_{c_{1}-m_{2}}^{N}\textbf{(}E_{c_{2}-m_{1}}^{N}\text{)}$ which attains
maximum value either when both the cavities are resonant with driving field,
i.e. $\Delta_{1}=\Delta_{2}=0$ or resonant with red sideband regime, i.e.
$\Delta_{1}=\Delta_{2}=-\omega_{b}$. Moreover when both the cavities are kept
in resonant with this red sideband regime the bipartite entanglement
$E_{c_{1}-b_{2}}^{N}\textbf{(}E_{c_{2}-b_{1}}^{N}\textbf{)}$ attains its
maximum value as shown in Fig. 2(c). Furthermore, it can be seen that if
cavity detunigs for both the cavities are kept fixed and resonant with blue
sideband regime i.e. $\Delta_{1}=\Delta_{2}=\omega_{b}$ then all the above
mentioned bipartite entanglements have significant values on gradually varying
both $\Delta_{m_{1}}/\omega_{b}$ and $\Delta_{m_{2}}/\omega_{b}$ from 0.8 to
1.1 as shown in Fig. 2(d)-2(f).

Figure 3: (Color Online) Density plot of bipartite entanglement in (a),(d),(g)
$E_{c_{1}-c_{2}}^{N}$; in (b),(e),(h)
$E_{c_{1}-m_{2}}^{N}=E_{c_{2}-m_{1}}^{N}$ and in (c),(f),(i)
$E_{c_{1}-b_{2}}^{N}=E_{c_{2}-b_{1}}^{N}$ versus detunings
$\Delta_{1}/\omega_{b}$ and $\Delta_{2}/\omega_{b}$. Here we have taken
$\Gamma=0.5\omega_{b}$ for (a),(b),(c); $\Gamma=0.8\omega_{b}$ for (d),(e),(f)
and $\Gamma=\omega_{b}$ for (g),(h),(i). We take both magnon detunings at
$\Delta_{m_{1}}=\Delta_{m_{2}}=\omega_{b}$ The other parameters are given in
Table 1

We plot five different distant bipartite entanglements as a function of
$\Delta_{1}/\omega_{b}$ and $\Delta_{2}/\omega_{b}$ for different photon
hopping factor $\Gamma$ as well while keepin both the magnon detuning in
resonant with blue sideband regime i.e.
$\Delta_{m_{1}}=\Delta_{m_{2}}=\omega_{b}$ in Fig. 3. For
$\Gamma==0.5\omega_{b}$ the quantity $E_{c_{1}-c_{2}}^{N}$ attains maximum
value when both the cavity have zero detunings i.e. $\Delta_{1}=\Delta_{2}=0$
whereas for off resonant cavities we get finite values of
$E_{c_{1}-c_{2}}^{N}$ as shown in Fig. 3(a). However the quantities
$E_{c_{1}-m_{2}}^{N}=E_{c_{2}-m_{1}}^{N}$ become maximum for two values of
cavity detunings which are $\Delta_{1}=\Delta_{2}=0$ and
$\Delta_{1}=\Delta_{2}=-\omega_{b}$ as shown in Fig. 3(b) whereas both the
quantities $E_{c_{1}-b_{2}}^{N}=E_{c_{2}-b_{1}}^{N}$ become maximum at
$\Delta_{1}=\Delta_{2}=-\omega_{b}$ as shown in Fig. 3(c). It can be seen that
if we increase photon hopping factor upto $\Gamma==0.8\omega_{b}$ then the
bipartite entanglement in between both the cavity modes $E_{c_{1}-c_{2}}^{N}$
becomes maximum for two cases i.e. for resonant cavities
$\Delta_{1}=\Delta_{2}=0$ and when resonant with red sideband regime
$\Delta_{1}=\Delta_{2}=-\omega_{b}$ as shown in Fig. 3(d). In addition, in the
density plots of the quantities $E_{c_{1}-m_{2}}^{N}=E_{c_{2}-m_{1}}^{N}$ the
panel corresponding to red sideband regime start to decrease whereas the panel
corresponding to resonant cavities increases as shown in Fig. 3(e). Moreover,
the quantities $E_{c_{1}-b_{2}}^{N}=E_{c_{2}-b_{1}}^{N}$ show the finite
values for a broad range of cavity detunings and attain maximum value for
$\Delta_{1}=\Delta_{2}=-1.5\omega_{b}$ as shown in Fig. 3(f). On further
increasing the value of $\Gamma$ and keeping it at $\Gamma==\omega_{b}$, the
quantity $E_{c_{1}-c_{2}}^{N}$ again becomes maximum for two cases i.e. for
$\Delta_{1}=\Delta_{2}=-0.5\omega_{b}$ and
$\Delta_{1}=\Delta_{2}=-1.5\omega_{b}$ as shown in Fig. 3(g) whereas the
quantities $E_{c_{1}-m_{2}}^{N}=E_{c_{2}-m_{1}}^{N}$ attain maximum value only
when both the cavity detunings are nearly resonant with blue sideband regime
as given in Fig. 3(h). However the quantities
$E_{c_{1}-b_{2}}^{N}=E_{c_{2}-b_{1}}^{N}$ attain maximum value only for very
far off-resonant cavities $\Delta_{1}=\Delta_{2}=-2\omega_{b}$ whereas for a
broad range of negative cavity detunings both these distant entanglements
almost become negligible however for a positive value of
$\Delta_{1}/\omega_{b}$ and $\Delta_{2}/\omega_{b}$ both the bipartite
entanglements attain finite values $E_{c_{1}-b_{2}}^{N}=E_{c_{2}-b_{1}}^{N}$
as shown in Fig. 3(i).

Figure 4: (Color Online) Density plot of bipartite entanglement in (a),(f)
$E^{N}_{c_{1}-c_{2}}$; in (b),(g) $E^{N}_{c_{1}-m_{2}}$; in (c),(h)
$E^{N}_{c_{2}-m_{1}}$; in (d),(i) $E^{N}_{c_{1}-b_{2}}$ and (e),(j)
$E^{N}_{c_{2}-b_{1}}$ versus $\Delta_{1}/\omega_{b}$ and $\Gamma/\omega_{b}$
for $\Delta_{2}=\omega_{b}$ in (a)-(e) and for $\Delta_{2}=-\omega_{b}$ in
(f)-(j). The other parameters are same as in Fig. 3.

We study the effects of varying photon hopping factor $\Gamma/\kappa_{c}$ and
normalised cavity detuning $\Delta_{1}/\omega_{b}$ on these five bipartite
entanglements while keeping second cavity detuning $\Delta_{2}/\omega_{b}$
fixed in Fig. 4. It can be seen that for $\Delta_{2}=\omega_{b}$ i.e. when
second cavity detuning is resonant with blue sideband regime, the quantity
$E_{c_{1}-c_{2}}^{N}$ becomes maximum for $\Delta_{1}$ varying in the range of
$0--0.5\omega_{b}$ whereas photon hopping factor varies upto $0-5$ although
after this range $E_{c_{1}-b_{2}}^{N}=E_{c_{2}-b_{1}}^{N}$ get finite value
for both positive and negative $\Delta_{1}/\omega_{b}$ with varying
$\Gamma\kappa_{c}$ as shown in Fig. 4(a). Similarly both the quantities
$E^{N}_{c_{1}-m_{2}}$ and $E^{N}_{c_{2}-m_{1}}$ get maximum for
$\Delta_{1}\approx-\omega_{b}$ and after this they attain finite values again
on varying $\Delta_{1}/\omega_{b}$ and $\Gamma\kappa_{c}$ as shown in Fig.
4(b) and 4(c). Moreover, the other two quantities $E_{c_{1}-b_{2}}^{N}$ and
$E_{c_{2}-b_{1}}^{N}$ attain their maximum value for $\Delta_{1}/\omega_{b}$
varying in the range of $(-1)$ to $(-2)$ even for a very high value of
$\Gamma\kappa_{c}$ as shown in Fig. 4 (d) and 4(e). In another scenario for
$\Delta_{2}=-\omega_{b}$ i.e. when second cavity detuning is resonant with red
sideband regime, the quantity $E_{c_{1}-c_{2}}^{N}$ becomes maximum nearby to
$\Delta_{1}/\omega_{b}=0$ and $\Gamma\approx\kappa_{c}$ however after this it
decreases very rapidly on gradually increasing $\Delta_{1}/\omega_{b}$ as well
as $\Gamma/\kappa_{c}$ as shown in Fig. 4(f). For this value of second cavity
detuning, it can be seen that both the bipartite entanglements
$E^{N}_{c_{1}-m_{2}}$ as well as $E^{N}_{c_{2}-m_{1}}$ get maximum only around
$\Delta_{1}/\omega_{b}=0$ to $\pm 1.0$ and $\Gamma/\kappa_{c}$ value lies in
between 7-10 and afterwards both these entanglements vanish as shown in Figs.
4(g) and 4(h). However in this range of $\Delta_{1}/\omega_{b}$,
$E_{c_{1}-b_{2}}^{N}$ and $E_{c_{2}-b_{1}}^{N}$ first become maximum for
single photon hopping factor $\Gamma/\kappa_{c}$ values which are in between
the range 5-7 and then both the bipartite entanglements become zero although a
further increase in $\Gamma/\kappa_{c}$ give maximum values of
$E_{c_{1}-b_{2}}^{N}$ and $E_{c_{2}-b_{1}}^{N}$ as depicted in Figs. 4(i) and
4(j).

Figure 5: (Color Online) Plot of different bipartite entanglement versus
$\Delta/\omega_{b}$ by taking $\Delta=\Delta_{1}=\Delta_{2}=\omega_{b}$ for
the upper panel and $\Delta=\Delta_{1}=-\Delta_{2}=\omega_{b}$ for the lower
panel. The other parameters are as in Fig. 3.

Furthermore, in Fig. 5 we plot different distant bipartite entanglements as a
function of $\Delta/\omega_{b}$ for symmetric cavities where we take
$\Delta=\Delta_{1}=\Delta_{2}=\omega_{b}$ (upper panel) and for antisymmetric
cavities $\Delta=\Delta_{1}=-\Delta_{2}=\omega_{b}$ (lower panel). For
$\Gamma=0.5\omega_{b}$ and symmetric cavities, the bipartite quantity
$E_{c_{1}-c_{2}}^{N}$ varies from 0 to 0.6 with gradually changing the
normalised detuning $\Delta/\omega_{b}$ in between -1 to 1 and after this
range the quantity $E_{c_{1}-c_{2}}^{N}$ becomes zero as shown in Fig. 5(a).
However, the other bipartite quantities $E_{m_{2}-b_{2}}^{N}$
$(E_{m_{1}-b_{1}}^{N})$ varies from 0 to 0.2 for negative values of
$\Delta/\omega_{b}$ and for $\Delta/\omega_{b}$ greater than zero both of
these quantities get saturated to a finite positive value. Moreover, the other
bipartite quantities $E_{c_{1}-b_{2}}^{N}$ ($E_{c_{2}-b_{1}}^{N}$) as well as
$E_{c_{1}-m_{2}}^{N}$ ($E_{c_{2}-m_{1}}^{N}$) become almost zero for positive
values of $\Delta/\omega_{b}$ as shown in Fig. 5(a). It can be seen that for
this value of $\Gamma$ a significant amount of entanglement transfer takes
place from $E_{m_{2}-b_{2}}^{N}$ $(E_{m_{1}-b_{1}}^{N})$ to
$E_{c_{1}-b_{2}}^{N}$ ($E_{c_{2}-b_{1}}^{N}$) and $E_{c_{1}-m_{2}}^{N}$
($E_{c_{2}-m_{1}}^{N}$) at $\Delta/\omega_{b}$ $\approx$ -0.3 and -1.2. For
$\Gamma=0.8\omega_{b}$, the bipartite entanglement $E_{c_{1}-c_{2}}^{N}$
become finite for $\Delta/\omega_{b}$ varying in the range of (0.3)-(1.3) and
(-0.5)-(-1.3) as shown in Fig. 5(b). It can be also seen that the bipartite
quantities $E_{m_{2}-b_{2}}^{N}$ $(E_{m_{1}-b_{1}}^{N})$ almost get around 0.2
for positive as well as negative values of $\Delta/\omega_{b}$ except for
certain values of $\Delta/\omega_{b}\approx 0.2$ and $-1.5$ whereas
$E_{c_{1}-m_{2}}^{N}$ ($E_{c_{2}-m_{1}}^{N}$) has finite values upto
$\Delta/\omega_{b}\approx 0.5$ and $E_{c_{1}-b_{2}}^{N}$
($E_{c_{2}-b_{1}}^{N}$) becomes zero even for negative values of
$\Delta/\omega_{b}$. In this case we get maximum entanglement transfer from
$E_{m_{2}-b_{2}}^{N}$ $(E_{m_{1}-b_{1}}^{N})$ to $E_{c_{1}-b_{2}}^{N}$
($E_{c_{2}-b_{1}}^{N}$) and $E_{c_{1}-m_{2}}^{N}$ ($E_{c_{2}-m_{1}}^{N}$)
around $\Delta/\omega_{b}$ $\approx$ 0.1 and -1.5. If we increase further
single photon hopping factor upto $\Gamma=\omega_{b}$ then the quantity
$E_{c_{1}-c_{2}}^{N}$ remains finite only for $\Delta/\omega_{b}$ varying in
the range of $(0.5)-(1.5)$ as well as $(-0.5)-(-1.5)$ whereas
$E_{m_{2}-b_{2}}^{N}$ $(E_{m_{1}-b_{1}}^{N})$ almost gets around 0.25 except
at certain values of the $\Delta/\omega_{b}$ for which the entanglement
transfer takes place between different bipartite correlations as shown in Fig.
5(c). In this case $E_{c_{1}-m_{2}}^{N}$ ($E_{c_{2}-m_{1}}^{N}$) has finite
values upto $\Delta/\omega_{b}\approx 1.0$ however $E_{c_{1}-b_{2}}^{N}$
($E_{c_{2}-b_{1}}^{N}$) qualitatively remains the same as depicted in Fig.
5(c). Moreover, the maximum entanglement transfer from $E_{m_{2}-b_{2}}^{N}$
$(E_{m_{1}-b_{1}}^{N})$ to $E_{c_{1}-b_{2}}^{N}$ ($E_{c_{2}-b_{1}}^{N}$) and
$E_{c_{1}-m_{2}}^{N}$ ($E_{c_{2}-m_{1}}^{N}$) takes place around values
$\Delta/\omega_{b}$ $\approx$ 0.5 and -1.8. Now for antisymmetric cavities
$\Delta=\Delta_{1}=-\Delta_{2}=\omega_{b}$ and $\Gamma=0.5\omega_{b}$ it can
be seen that both the bipartite entanglements $E_{m_{2}-b_{2}}^{N}$
$(E_{m_{1}-b_{1}}^{N})$ have finite values with a varying $\Delta/\omega_{b}$
although for few values both becomes zero s shown in Fig. 5(d). All other
bipartite entanglements have very small values for this value of $\Gamma$. For
$\Gamma=0.8\omega_{b}$ the bipartite entanglement $E_{c_{1}-c_{2}}^{N}$
becomes zero whereas the quantities $E_{m_{2}-b_{2}}^{N}$
$(E_{m_{1}-b_{1}}^{N})$ have finite values from 0.1-0.25 as shown in Fig.
5(d). Moreover, the bipartite entanglements $E_{c_{1}-b_{2}}^{N}$
($E_{c_{2}-b_{1}}^{N}$) increases for this value of $\Gamma$ and become finite
for a varying $\Delta/\omega_{b}$ in between the range of (-1)-(1) whereas
$E_{c_{1}-m_{2}}^{N}$ ($E_{c_{2}-m_{1}}^{N}$) also increases and varies from
0-0.07(0.08) with $\Delta/\omega_{b}$ as depicted in Fig. 5(d). With a further
increment in $\Gamma$ both the bipartite entanglements $E_{c_{1}-m_{2}}^{N}$
($E_{c_{2}-m_{1}}^{N}$) becomes finite over whole range of varying
$\Delta/\omega_{b}$ whereas all other bipartite entanglements qualitatively
remains the same (like earlier case of $\Gamma=0.8\omega_{b}$) as shown in
Fig. 5(f).

##  V Conclusions

We present an experimentally feasible scheme based on coupled magnomechanical
system where two microwave cavities are coupled through single photon hopping
parameter $\Gamma$ and each cavity also contains a magnon mode and phonon
mode. We have investigated continuous variable entanglement between distant
bipartitions for an appropriate set of both cavities and magnons detuning and
their decay rates. Hence, it can be seen that bipartite entanglement between
indirectly coupled systems are substantial in our proposed scheme. Moreover,
cavity-cavity coupling strength also plays a key role in the degree of
bipartite entanglement and its transfer among different direct and indirect
modes. This scheme may prove to be significant for processing continuous
variable quantum information in quantum memory protocols.

## Appendix A. Stability of the system

It is worthy to discuss the stability of the subject system, since the
stability of strongly magnomechanical system is difficult to achieve. In this
section, we are going to discuss the stability of our system. The system
becomes stable only when all the eigenvalues of the drift matrix $\mathcal{M}$
have negative real parts. In case the sign of the real part of eigenvalues
changes, system will becomes unstable. Hence, in order to provide the
intuitive picture for the parameter regime where stability occurs can be
obtained from the Routh-Hurwitz criterion, we have plotted the maximum of the
real parts of the eigenvalues $\lambda_{\mathcal{M}}$
($|\mathcal{M}-\lambda_{\mathcal{M}}\mathbb{1}|=0$) of the drift matrix
$\mathcal{M}$ vs the normalized detunings in Fig. 7 [70]. It is important to
mention here that the system becomes unstable if the maximum of the real parts
of the eigenvalues $\lambda_{\mathcal{M}}$ is greater than zero. Figure. 7
clearly shows the maximum of the real parts of the eigenvalues
$\lambda_{\mathcal{M}}$ remains negative for the chosen parameters and hence
the system is stable. Therefore, the whole set of numerical parameters used
throughout the manuscript satisfies the stability conditions and hence the
working regime we chose is the regime of stability.

## Data availability

All data used during this study are available within the article.

## References

## References

  * Schwabl [2007] F. Schwabl,  _Quantum mechanics_ (Springer Science & Business Media, 2007).
  * Ballentine [2014] L. E. Ballentine,  _Quantum mechanics: a modern development_ (World Scientific Publishing Company, 2014).
  * Horodecki  _et al._ [2009] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Reviews of modern physics 81, 865 (2009).
  * Vitali  _et al._ [2007] D. Vitali, S. Gigan, A. Ferreira, H. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, Optomechanical entanglement between a movable mirror and a cavity field, Physical review letters 98, 030405 (2007).
  * Mancini  _et al._ [2002] S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, Entangling macroscopic oscillators exploiting radiation pressure, Physical review letters 88, 120401 (2002).
  * Yang  _et al._ [2015] C.-J. Yang, J.-H. An, W. Yang, and Y. Li, Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering, Physical Review A 92, 062311 (2015).
  * Li  _et al._ [2017] J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, Enhanced entanglement of two different mechanical resonators via coherent feedback, Physical Review A 95, 043819 (2017).
  * Hartmann and Plenio [2008] M. J. Hartmann and M. B. Plenio, Steady state entanglement in the mechanical vibrations of two dielectric membranes, Physical Review Letters 101, 200503 (2008).
  * Liao  _et al._ [2014] J.-Q. Liao, Q.-Q. Wu, F. Nori, _et al._ , Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system, Physical Review A 89, 014302 (2014).
  * Xiong  _et al._ [2005] H. Xiong, M. O. Scully, and M. S. Zubairy, Correlated spontaneous emission laser as an entanglement amplifier, Physical review letters 94, 023601 (2005).
  * Kiffner  _et al._ [2007] M. Kiffner, M. S. Zubairy, J. Evers, and C. Keitel, Two-mode single-atom laser as a source of entangled light, Physical Review A 75, 033816 (2007).
  * Qamar  _et al._ [2009] S. Qamar, M. Al-Amri, S. Qamar, and M. S. Zubairy, Entangled radiation via a raman-driven quantum-beat laser, Physical Review A 80, 033818 (2009).
  * Chen  _et al._ [2019] Z. Chen, J.-X. Peng, J.-J. Fu, and X.-L. Feng, Entanglement of two rotating mirrors coupled to a single laguerre-gaussian cavity mode, Optics express 27, 29479 (2019).
  * Bhattacharya  _et al._ [2008] M. Bhattacharya, P.-L. Giscard, and P. Meystre, Entanglement of a laguerre-gaussian cavity mode with a rotating mirror, Physical Review A 77, 013827 (2008).
  * Peng  _et al._ [2019] J.-X. Peng, Z. Chen, Q.-Z. Yuan, and X.-L. Feng, Optomechanically induced transparency in a laguerre-gaussian rotational-cavity system and its application to the detection of orbital angular momentum of light fields, Physical Review A 99, 043817 (2019).
  * Singh  _et al._ [2021] S. Singh, J.-X. Peng, M. Asjad, and M. Mazaheri, Entanglement and coherence in a hybrid laguerre–gaussian rotating cavity optomechanical system with two-level atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 54, 215502 (2021).
  * Cheng  _et al._ [2021] H.-J. Cheng, S.-J. Zhou, J.-X. Peng, A. Kundu, H.-X. Li, L. Jin, and X.-L. Feng, Tripartite entanglement in a laguerre–gaussian rotational-cavity system with an yttrium iron garnet sphere, JOSA B 38, 285 (2021).
  * Rameshti  _et al._ [2022] B. Z. Rameshti, S. V. Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, Cavity magnonics, Physics Reports 979, 1 (2022).
  * Rao  _et al._ [2021] J. Rao, P. Xu, Y. Gui, Y. Wang, Y. Yang, B. Yao, J. Dietrich, G. Bridges, X. Fan, D. Xue, _et al._ , Interferometric control of magnon-induced nearly perfect absorption in cavity magnonics, Nature communications 12, 1933 (2021).
  * Zhang  _et al._ [2021] G.-Q. Zhang, Z. Chen, W. Xiong, C.-H. Lam, and J. You, Parity-symmetry-breaking quantum phase transition via parametric drive in a cavity magnonic system, Physical Review B 104, 064423 (2021).
  * Shen  _et al._ [2021] R.-C. Shen, Y.-P. Wang, J. Li, S.-Y. Zhu, G. Agarwal, and J. You, Long-time memory and ternary logic gate using a multistable cavity magnonic system, Physical Review Letters 127, 183202 (2021).
  * Tabuchi  _et al._ [2014a] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Hybridizing ferromagnetic magnons and microwave photons in the quantum limit, Physical review letters 113, 083603 (2014a).
  * Zhang  _et al._ [2014a] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Strongly coupled magnons and cavity microwave photons, Physical review letters 113, 156401 (2014a).
  * Kittel [1948a] C. Kittel, On the theory of ferromagnetic resonance absorption, Physical review 73, 155 (1948a).
  * Li  _et al._ [2023] J. Li, Y.-P. Wang, J.-Q. You, and S.-Y. Zhu, Squeezing microwaves by magnetostriction, National Science Review 10, nwac247 (2023).
  * Li  _et al._ [2018] J. Li, S.-Y. Zhu, and G. Agarwal, Magnon-photon-phonon entanglement in cavity magnomechanics, Physical review letters 121, 203601 (2018).
  * Ullah  _et al._ [2020] K. Ullah, M. T. Naseem, and Ö. E. Müstecaplıoğlu, Tunable multiwindow magnomechanically induced transparency, fano resonances, and slow-to-fast light conversion, Physical Review A 102, 033721 (2020).
  * Sohail  _et al._ [2023a] A. Sohail, R. Ahmed, J.-X. Peng, T. Munir, A. Shahzad, S. Singh, and M. C. de Oliveira, Controllable fano-type optical response and four-wave mixing via magnetoelastic coupling in an opto-magnomechanical system, Journal of Applied Physics 133 (2023a).
  * Liu  _et al._ [2023] Z.-X. Liu, J. Peng, and H. Xiong, Generation of magnonic frequency combs via a two-tone microwave drive, Physical Review A 107, 053708 (2023).
  * Singh  _et al._ [2023] S. Singh, M. Mazaheri, J.-X. Peng, A. Sohail, M. Khalid, and M. Asjad, Enhanced weak force sensing based on atom-based coherent quantum noise cancellation in a hybrid cavity optomechanical system, Frontiers in Physics 11, 245 (2023).
  * Amazioug  _et al._ [2023] M. Amazioug, B. Teklu, and M. Asjad, Enhancement of magnon–photon–phonon entanglement in a cavity magnomechanics with coherent feedback loop, Scientific Reports 13, 3833 (2023).
  * Zhang  _et al._ [2015] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Magnon dark modes and gradient memory, Nature communications 6, 8914 (2015).
  * Shen  _et al._ [2022] R.-C. Shen, J. Li, Z.-Y. Fan, Y.-P. Wang, and J. You, Mechanical bistability in kerr-modified cavity magnomechanics, Physical Review Letters 129, 123601 (2022).
  * Wang  _et al._ [2018] Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. You, Bistability of cavity magnon polaritons, Physical review letters 120, 057202 (2018).
  * Kong  _et al._ [2019] C. Kong, H. Xiong, and Y. Wu, Magnon-induced nonreciprocity based on the magnon kerr effect, Physical Review Applied 12, 034001 (2019).
  * Zhang  _et al._ [2023] G.-Q. Zhang, Y. Wang, and W. Xiong, Detection sensitivity enhancement of magnon kerr nonlinearity in cavity magnonics induced by coherent perfect absorption, Physical Review B 107, 064417 (2023).
  * Xiong  _et al._ [2022] W. Xiong, M. Tian, G.-Q. Zhang, and J. You, Strong long-range spin-spin coupling via a kerr magnon interface, Physical Review B 105, 245310 (2022).
  * Wang  _et al._ [2016] Y.-P. Wang, G.-Q. Zhang, D. Zhang, X.-Q. Luo, W. Xiong, S.-P. Wang, T.-F. Li, C.-M. Hu, and J. You, Magnon kerr effect in a strongly coupled cavity-magnon system, Physical Review B 94, 224410 (2016).
  * Zhang  _et al._ [2019a] G. Zhang, Y. Wang, and J. You, Theory of the magnon kerr effect in cavity magnonics, Science China Physics, Mechanics & Astronomy 62, 1 (2019a).
  * Hisatomi  _et al._ [2016] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Bidirectional conversion between microwave and light via ferromagnetic magnons, Physical Review B 93, 174427 (2016).
  * Yan  _et al._ [2020] Z. Yan, C. Wan, and X. Han, Magnon blocking effect in an antiferromagnet-spaced magnon junction, Physical Review Applied 14, 044053 (2020).
  * Liu  _et al._ [2019] Z.-X. Liu, H. Xiong, and Y. Wu, Magnon blockade in a hybrid ferromagnet-superconductor quantum system, Physical Review B 100, 134421 (2019).
  * Yu  _et al._ [2020] M. Yu, H. Shen, and J. Li, Magnetostrictively induced stationary entanglement between two microwave fields, Physical Review Letters 124, 213604 (2020).
  * Hidki  _et al._ [2023a] A. Hidki, A. Lakhfif, J. El Qars, and M. Nassik, Evolution of rényi-2 quantum correlations in a double cavity–magnon system, Modern Physics Letters A , 2350044 (2023a).
  * Li and Zhu [2019] J. Li and S.-Y. Zhu, Entangling two magnon modes via magnetostrictive interaction, New Journal of Physics 21, 085001 (2019).
  * Zhang  _et al._ [2019b] Z. Zhang, M. O. Scully, and G. S. Agarwal, Quantum entanglement between two magnon modes via kerr nonlinearity driven far from equilibrium, Physical Review Research 1, 023021 (2019b).
  * Yang  _et al._ [2020] Z.-B. Yang, J.-S. Liu, H. Jin, Q.-H. Zhu, A.-D. Zhu, H.-Y. Liu, Y. Ming, and R.-C. Yang, Entanglement enhanced by kerr nonlinearity in a cavity optomagnonics system, Optics Express 28, 31862 (2020).
  * Hussain  _et al._ [2022] B. Hussain, S. Qamar, and M. Irfan, Entanglement enhancement in cavity magnomechanics by an optical parametric amplifier, Physical Review A 105, 063704 (2022).
  * Hidki  _et al._ [2023b] A. Hidki, Y.-L. Ren, A. Lakhfif, J. El Qars, and M. Nassik, Enhanced maximum entanglement between two microwave fields in the cavity magnomechanics with an optical parametric amplifier, Physics Letters A 463, 128667 (2023b).
  * Sohail  _et al._ [2023b] A. Sohail, R. Ahmed, J.-X. Peng, A. Shahzad, and S. Singh, Enhanced entanglement via magnon squeezing in a two-cavity magnomechanical system, JOSA B 40, 1359 (2023b).
  * Hidki  _et al._ [2023c] A. Hidki, A. Lakhfif, J. El Qars, and M. Nassik, Transfer of squeezing in a cavity magnomechanics system, Journal of Modern Optics , 1 (2023c).
  * Wu  _et al._ [2021] W.-J. Wu, Y.-P. Wang, J.-Z. Wu, J. Li, and J. You, Remote magnon entanglement between two massive ferrimagnetic spheres via cavity optomagnonics, Physical Review A 104, 023711 (2021).
  * Sun  _et al._ [2021] F.-X. Sun, S.-S. Zheng, Y. Xiao, Q. Gong, Q. He, and K. Xia, Remote generation of magnon schrödinger cat state via magnon-photon entanglement, Physical Review Letters 127, 087203 (2021).
  * Xie  _et al._ [2023] H. Xie, L.-W. He, C.-G. Liao, Z.-H. Chen, and X.-M. Lin, Generation of robust optical entanglement in cavity optomagnonics, Optics Express 31, 7994 (2023).
  * Chen  _et al._ [2021] Y.-T. Chen, L. Du, Y. Zhang, and J.-H. Wu, Perfect transfer of enhanced entanglement and asymmetric steering in a cavity-magnomechanical system, Physical Review A 103, 053712 (2021).
  * Dilawaiz and Irfan [2022] S. Q. Dilawaiz and M. Irfan, Entangled atomic ensemble and yig sphere in coupled microwave cavities, arXiv preprint arXiv:2211.14914 (2022).
  * Zhang  _et al._ [2016] X. Zhang, C. Zou, L. Jiang, and H. Tang, Cavity magnomechanics sci, Adv 2, e1501286 (2016).
  * Li  _et al._ [2019] J. Li, S.-Y. Zhu, and G. Agarwal, Squeezed states of magnons and phonons in cavity magnomechanics, Physical Review A 99, 021801 (2019).
  * Kittel [1958] C. Kittel, Interaction of spin waves and ultrasonic waves in ferromagnetic crystals, Physical Review 110, 836 (1958).
  * Zhang  _et al._ [2014b] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Strongly coupled magnons and cavity microwave photons, Physical review letters 113, 156401 (2014b).
  * Gonzalez-Ballestero  _et al._ [2020] C. Gonzalez-Ballestero, D. Hümmer, J. Gieseler, and O. Romero-Isart, Theory of quantum acoustomagnonics and acoustomechanics with a micromagnet, Physical Review B 101, 125404 (2020).
  * Osada  _et al._ [2016] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Cavity optomagnonics with spin-orbit coupled photons, Physical review letters 116, 223601 (2016).
  * Kittel [1948b] C. Kittel, On the theory of ferromagnetic resonance absorption, Physical review 73, 155 (1948b).
  * Tabuchi  _et al._ [2014b] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Hybridizing ferromagnetic magnons and microwave photons in the quantum limit, Physical review letters 113, 083603 (2014b).
  * Huebl  _et al._ [2013] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. Goennenwein, High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids, Physical Review Letters 111, 127003 (2013).
  * Goryachev  _et al._ [2014] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, High-cooperativity cavity qed with magnons at microwave frequencies, Physical Review Applied 2, 054002 (2014).
  * Simon [2000] R. Simon, Peres-horodecki separability criterion for continuous variable systems, Physical Review Letters 84, 2726 (2000).
  * Holstein and Primakoff [1940] T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Physical Review 58, 1098 (1940).
  * Gardiner and Zoller [2000] C. W. Gardiner and P. Zoller, Quantum noise, vol. 56 of springer series in synergetics, Springer–Verlag, Berlin 97, 98 (2000).
  * DeJesus and Kaufman [1987] E. X. DeJesus and C. Kaufman, Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Physical Review A 35, 5288 (1987).
  * Parks and Hahn [1993] P. C. Parks and V. Hahn,  _Stability theory_ (Prentice-Hall, Inc., 1993).
  * Sohail  _et al._ [2020] A. Sohail, M. Rana, S. Ikram, T. Munir, T. Hussain, R. Ahmed, and C.-s. Yu, Enhancement of mechanical entanglement in hybrid optomechanical system, Quantum Information Processing 19, 1 (2020).
  * Eisert [2001] J. Eisert,  _Entanglement in quantum theory_ , Ph.D. thesis, Ph. D. Thesis, University of Potsdam, Postdam, Germany (2001).
  * Vidal and Werner [2002] G. Vidal and R. F. Werner, Computable measure of entanglement, Physical Review A 65, 032314 (2002).
  * Plenio [2005] M. B. Plenio, Logarithmic negativity: a full entanglement monotone that is not convex, Physical review letters 95, 090503 (2005).
  * Adesso and Illuminati [2007] G. Adesso and F. Illuminati, Entanglement in continuous-variable systems: recent advances and current perspectives, Journal of Physics A: Mathematical and Theoretical 40, 7821 (2007).