# Nonreciprocal high-order sidebands induced by magnon Kerr nonlinearity

Mei Wang<EMAIL_ADDRESS>School of Electrical Engineering, Wuhan
Polytechnic University, Wuhan, 430040, the People’s Republic of China  Cui
Kong  College of Physics and Electronic Science, Hubei Normal University,
Huangshi, 435002, the People’s Republic of China  Zhao-Yu Sun  Department of
mathematics and Physics, Wuhan Polytechnic University, Wuhan, 430040, the
People’s Republic of China  Duo Zhang  Department of mathematics and Physics,
Wuhan Polytechnic University, Wuhan, 430040, the People’s Republic of China
Yu-Ying Wu  Department of mathematics and Physics, Wuhan Polytechnic
University, Wuhan, 430040, the People’s Republic of China  Li-Li Zheng
<EMAIL_ADDRESS>Key Laboratory of Optoelectronic Chemical Materials and
Devices of Ministry of Education, Jianghan University, Wuhan, 430074, People’s
Republic of China

###### Abstract

We propose an effective approach for creating robust nonreciprocity of high-
order sidebands, including the first-, second- and third-order sidebands, at
microwave frequencies. This approach relies on magnon Kerr nonlinearity in a
cavity magnonics system composed of two microwave cavities and one yttrium
iron garnet (YIG) sphere. By manipulating the driving power applied on YIG and
the frequency detuning between the magnon mode in YIG and the driving field,
the effective Kerr nonlinearity can be strengthened, thereby inducing strong
transmission non-reciprocity. More interestingly, we find the higher the
sideband order, the stronger the transmission nonreciprocity marked by the
higher isolation ratio in the optimal detuning regime. Such a series of
equally-spaced high-order sidebands have potential applications in frequency
comb-like precision measurement, besides structuring high-performance on-chip
nonreciprocal devices.

###### pacs:

42.25.Bs, 03.65.Ta

##  I Introduction

Recently years, cavity magnonics system, benefited from its unique features,
has shown potential application prospects in information processing.
Especially, the vast spin excitation in YIG sphere called magnon, which plays
a vital role in the system. Benefited from the high spin density, strong and
ultra-strong magnon-microwave photon coupling Tobar2014 ; Bauer2019 ;
Ruoso2017 ; Tang2014 ; Liu2019 ; Flatt2010 , observation of exceptional point
and precision magnetometry Ruoso2020 ; Yan2019  have been demonstrated in
experiment. Furthermore, thanks to the high tunability, extensibility and low
dissipation of the magnon, complex information interaction platforms
Nakamura2019  can be structured via establishing the interaction of magnon to
an (artificial) atom Nakamura2015 , acoustic phonons Agarwals2019 ; Jaswa1998
; Tang2016 ; Wang2017 ; Wang2020 , optical photons Marquardt2016 ;
Ferguson2015 , or superconducting coplanar microwave resonator Goennenwein2013
. In addition to above merits, another advantage of the system is the Kerr
nonlinearity of magnon originated from the magnetocrystalline anisotropy in
YIG sphere. Based on Kerr nonlinearity, a series of studies You2018 ; Kons2015
; Wu2019 ; Zhang2020  have been carried out, and one of the most notable is
the nonreciprocity.

Nonreciprocity, associated with the breaking of Lorentz reciprocity Potton2004
; Deck2018 , is manifested as asymmetric propagation of signal in two opposite
directions. Nonreciprocal physics has vast and valuable applications in
communication technology and information processing. For examples, one-way
signal communication and manufacturing chip-scale nonreciprocal devices:
isolator, circulator, diode. Stimulated by the bright prospect, considerable
efforts has been made in the study of nonreciprocal physics. One of the most
watched is the nonreciprocity of electromagnetic fields Caloz2018 ;
Asadchy2020 , which is mainly concerned with the nonreciprocal propagation of
electromagnetic fields in the microwave and optical regions. Microwave and
optical nonreciprocity are both typically achieved from the magneto-optic
crystal H2000 ; Ram2007 ; Raghu2008 ; Osgood2008 ; Solja2009  relying on
Faraday-rotation Fan2003 ; Assanto2008 ; Sai2013  to break the Lorentz
reciprocity. However, the tricky thing is the bulky and heavy structure of the
magneto-optic crystal, that poses a serious hurdle to flexibly control and
chip-scale integration. Over the past few decades, alternative mechanisms
towards realizing nonreciprocal propagation at microwave and optical
frequencies have been proposed. The most promising ones are based on
nonlinearity Review1 ; Review2 ; Review3 ; Review4 ; Review5 ; Review6 ;
Review7 ; Aumentado2017 ; Ustinov2012 ; Kippen2018 , reservoir engineering
Clerk2017 ; Clerk2015 , parametric time modulation Alu2017 ; Aumentado2014 .

Inspired by previous studies, based on nonlinearity mechanism, we propose a
flexible scheme to achieve robust nonreciprocity of the first-, second-, and
third-order sidebands at microwave frequencies. This approach is implemented
in a three-mode cavity magnonics system, which consists of two coupled
microwave cavities and a YIG (or another suitable magnonic material) sphere.
Due to the magnon Kerr nonlinearity in YIG sphere, strong nonreciprocity of
high-order sidebands can be obtained by flexibly manipulating the driving
power injected on YIG sphere and the frequency detuning between magnon mode
(in YIG) and the driving field. On one hand, the driving power applied on YIG
affects excitation of magnon that determines the effective strength of the
Kerr nonlinearity. It therefor induces varying degrees asymmetric responses of
the system in two opposite directions. On the other hand, frequency detuning
between the driving field and the magnon mode also affects magnon excitation
in YIG sphere. Thus the effective Kerr nonlinearity will be influenced, as
well as degree of the asymmetric responses in two directions. More
interestingly, in the optimal parameters regime, the greater the order of the
sideband, the stronger transmission nonreciprocity marked by the enhanced
isolation ratio. Such a series of equally-spaced high-order sidebands can be
applied in frequency comb-like precision measurement, Besides structuring
nonreciprocal devices. In addition, frequencies of the sidebands avoid the
main cavity frequency and this will strongly inhibit dissipation. The system
parameters are referenced from recent experiments You2018 ; Wang2016 ,
specially, Kerr nonlinear coefficient ($K$ reaches $10^{2}$ nHz refereing to
Zhang2019 ) in the system is highly dependent on the specific experimental
implementation. This study broadens applications of the cavity magnonics
system in information processing.

This paper is organized as follows: In Sec. II, we introduce theoretical
scheme of the three-mode cavity magnonics system and its effective Hamiltonian
with external pump fields. Then we derive transmission coefficients and
isolation ratios of the first-, second- and third-order sidebands in two
cases. In Sec. III, we study transmission nonreciprocity of the high-order
sidebands through manipulating the driving field on the YIG sphere and the
frequency detuning between driving field and the magnon mode in YIG sphere.
Conclusions are finally drawn in Sec. IV.

Figure 1: (Color online) (a) Schematic diagram of a three-mode cavity
magnonics system consisting of two coupled microwave cavities and a YIG
sphere. (b) Frequency spectrogram of the cavity magnonics system. Here the
driving field (with amplitude $\varepsilon$) applied on YIG sphere and the
control fields (with amplitudes $\varepsilon_{a}$, $\varepsilon_{b}$) injected
from microwave cavities have the same frequency $\omega_{d}$. Frequencies of
the control fields and the probe fields (with amplitudes $s_{a}$, $s_{b}$) are
respectively detuned from the cavities’ frequency by $\Delta_{c}$ and
$\delta$. And the frequency detuning between the probe field and the control
field is $\Delta_{p}$. There are higher-order sidebands at frequency
$\omega_{d}+n\Delta_{p}$, where $n$ is an integer representing the sideband
order.

##  II The Model and formula derivation

We consider a three-mode cavity magnonics system You2018  as shown in Fig.
1(a). It is composed of two three-dimensional rectangular microwave cavities
($a$ and $b$, frequency $\omega_{c}$) and one YIG sphere, where collective
spin excitation is quantified as magnon ($m$, frequency $\omega_{m}$). The
cavities are linearly coupled to each other with coupling rate $J$; and each
of them is driven and detected by a bichromatic microwave source, i.e., a
strong control field with amplitude $\varepsilon_{a}$ ($\varepsilon_{b}$) at
frequency $\omega_{d}$ and a weak probe field $s_{a}$ ($s_{b}$) at frequency
$\omega_{p}$. The YIG sphere, located in one cavity, is exposed to a uniformly
DC magnetic bias field (${H}_{z}$) in the $z$ direction. Meanwhile, it
interacts with one cavity with strength $g$, and is pumped by a microwave
driving field with amplitude $\varepsilon$ and frequency $\omega_{d}$. Note
that amplitudes of the input microwave fields can be converted into
corresponding powers, i.e., $P_{j}=\hbar\omega_{d}{\varepsilon}^{2}_{j}$,
$P=\hbar\omega_{d}{\varepsilon}^{2}$ and
$P_{s_{j}}=\hbar\omega_{p}{s}^{2}_{j}$ ($j=a,b$) with control power $P_{j}$,
driving power $P$ and probe power $P_{s_{j}}$.

In the rotating frame, including all external microwave sources, the system
Hamiltonian (refering to You2018 ; Wang2016 ) can be written as

$\displaystyle\hat{H}_{\rm tot}/\hbar$ $\displaystyle=$
$\displaystyle\Delta_{c}(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b})+\Delta_{m}\hat{m}^{\dagger}\hat{m}+K\hat{m}^{\dagger}\hat{m}\hat{m}^{\dagger}\hat{m}$
(1)
$\displaystyle+J(\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}^{\dagger})+g(\hat{b}^{\dagger}\hat{m}+\hat{b}\hat{m}^{\dagger})$
$\displaystyle+i\sqrt{\eta_{a}\kappa_{a}}\varepsilon_{a}(\hat{a}^{\dagger}-\hat{a})+i\sqrt{\eta_{b}\kappa_{b}}\varepsilon_{b}(\hat{b}^{\dagger}-\hat{b})$
$\displaystyle+i\sqrt{\eta_{a}\kappa_{a}}s_{a}(\hat{a}^{\dagger}e^{-i\Delta_{p}t}-\hat{a}e^{i\Delta_{p}t})$
$\displaystyle+i\sqrt{\eta_{b}\kappa_{b}}s_{b}(\hat{b}^{\dagger}e^{-i\Delta_{p}t}-\hat{b}e^{i\Delta_{p}t})$
$\displaystyle+i\sqrt{\eta_{m}\kappa_{m}}\varepsilon(\hat{m}^{\dagger}-\hat{m}),$

here $\Delta_{m}=\omega_{m}-\omega_{d}$, $\Delta_{p}=\omega_{p}-\omega_{c}$
and $\Delta_{c}=\omega_{c}-\omega_{d}$ is constant. In addition, the frequency
detunings satisfy relationship $\Delta_{p}=\Delta_{c}+\delta$, where
$\delta=\omega_{p}-\omega_{c}$ refereing to Fig. 1(b). The third term in
Hamiltonian is magnon Kerr nonlinear term originating from the
magnetocrystalline anisotropy in the YIG sphere. The Kerr nonlinear
coefficient $K=\mu_{0}K_{an}{\gamma}^{2}/(M^{2}V_{m})$, where $\mu_{0}$ is the
magnetic permeability of free space, $K_{an}$ is the first-order anisotropy
constant, $\gamma$ is the gyromagnetic ratio, $M$ is the saturation
magnetization, and $V_{m}$ is the volume of the YIG sphere. $\kappa_{j}$
($j=a,b,m$) represents the total dissipation of any microwave cavity or magnon
mode with cavity (magnon) coupling parameter $\eta_{j}$.

In this work, we employ analytic solution method to solve Heisenberg-Langevin
equations. It not only describes the dynamical evolution of the cavity
magnonics system, but also responses the expectation values of all operators,
namely, $\langle\hat{o}\rangle=o$ ($\hat{o}$ is a normal annihilation
operator). Taking account damping progresses in the system, Heisenberg-
Langevin equations of the system can be expressed as

$\displaystyle\dot{{a}}$ $\displaystyle=$
$\displaystyle-(i\Delta_{c}+\frac{\kappa_{a}}{2}){a}-iJ{b}+\sqrt{\eta_{a}\kappa_{a}}\varepsilon_{a}$
(2a) $\displaystyle+\sqrt{\eta_{a}\kappa_{a}}s_{a}e^{-i\Delta_{p}t},$
$\displaystyle\dot{{b}}$ $\displaystyle=$
$\displaystyle-(i\Delta_{c}+\frac{\kappa_{b}}{2}){b}-iJ{a}-ig{m}+\sqrt{\eta_{b}\kappa_{b}}\varepsilon_{b}$
(2b) $\displaystyle+\sqrt{\eta_{b}\kappa_{b}}s_{b}e^{-i\Delta_{p}t},$
$\displaystyle\dot{{m}}$ $\displaystyle=$
$\displaystyle-(i\Delta_{m}+\frac{\kappa_{m}}{2}){m}-ig{b}-i(2K|{m}|^{2}+K){m}$
(2c) $\displaystyle+\sqrt{\eta_{m}\kappa_{m}}\varepsilon.$

Equations (2a) and (2b) describe the dynamics of cavities $a$ and $b$.
Equation (2c) describes the dynamics of the magnon mode. The cubic term
$-2iK|{m}|^{2}{m}$ in Equation (2c) presents the nonlinearity of the system.
It is derived from the Kerr nonlinearity term in Hamiltonian (1) with
factorizing averages approximation, i.e., $\langle bc\rangle=\langle
b\rangle\langle c\rangle$. The input quantum and thermal noise terms are
ignored here because their mean values are zero.

For existence of the nonlinear term, equations (2a)-(2c) are inherently
nonlinear and cannot be solved directly and precisely. However, due to the
strength of the control fields ($5\,\,m$W) are far stronger than the probe
fields ($1\,\,n$W) in the system, we can take perturbation method

Figure 2: (Color online) The steady-state microwave photon numbers
$|\alpha|^{2}$, $|\beta|^{2}$ and magnon number $|\varpi|^{2}$ versus with the
resonant driving power $P$ on the magnon mode in two cases. (a) case 1:
$P_{a}=5\,\,m$W, $P_{s_{a}}=1\,\,n$W, $P_{b}=P_{s_{b}}=0$; (b) case $2$:
$P_{b}=5\,\,m$W, $P_{s_{b}}=1\,\,n$W, $P_{a}=P_{s_{a}}=0$. Other parameters
are given as $\omega_{c}/2\pi=10.1$ GHz,
$\omega_{m}/2\pi=\omega_{d}/2\pi=10.06$ GHz, $\Delta_{c}/2\pi=40$ MHz,
$\Delta_{m}=0$, $\kappa_{a}/2\pi=3.8$ MHz, $\kappa_{b}/2\pi=5$ MHz,
$\kappa_{m}/2\pi=20$ MHz, $\delta/2\pi=-12$ MHz, $K/2\pi=3.9\times 10^{-7}$
Hz, $J/2\pi=12$ MHz, $g/2\pi=8$ MHz.

to solve equations (2). That is dividing each operator into two parts: a
steady-state mean value and a perturbation term, i.e., $a={\alpha}+\delta a$,
$b={\beta}+\delta b$, $m={\varpi}+\delta m$. It means that the steady-state
mean values of the operators are determined by the strong control fields, and
the perturbation terms are caused by the weak probe fields. When the probe
fields are absent, with $d{\alpha}/dt=0$, $d{\beta}/dt=0$ and
$d{\varpi}/dt=0$, the mean values of the operators from the steady-state
equations can be derived as

$\displaystyle{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\eta_{a}\kappa_{a}}\varepsilon_{a}-iJ{\beta}}{i\Delta_{c}+\frac{\kappa_{a}}{2}},$
(3) $\displaystyle{\beta}$ $\displaystyle=$
$\displaystyle\frac{g(\Delta_{c}-i\frac{\kappa_{a}}{2}){\varpi}-iJ\sqrt{\eta_{a}\kappa_{a}}\varepsilon_{a}+(i\Delta_{c}+\frac{\kappa_{a}}{2})\sqrt{\eta_{b}\kappa_{b}}\varepsilon_{b}}{(i\Delta_{c}+\frac{\kappa_{a}}{2})(i\Delta_{c}+\frac{\kappa_{b}}{2})+J^{2}},$
$\displaystyle{\varpi}$ $\displaystyle=$
$\displaystyle\frac{-gJ\sqrt{\eta_{a}\kappa_{a}}\varepsilon_{a}+g(\Delta_{c}-i\frac{\kappa_{a}}{2})\sqrt{\eta_{b}\kappa_{b}}\varepsilon_{b}}{\left[(i\Delta_{c}+\frac{\kappa_{a}}{2})(i\Delta_{c}+\frac{\kappa_{b}}{2})+J^{2}\right]{\Theta}}+\frac{\sqrt{\eta_{m}\kappa_{m}}\varepsilon}{{\Theta}},$

where we have defined

$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\frac{g^{2}(i\Delta_{c}+\frac{\kappa_{a}}{2})}{(i\Delta_{c}+\frac{\kappa_{a}}{2})(i\Delta_{c}+\frac{\kappa_{b}}{2})+J^{2}}+\frac{\kappa_{m}}{2}$
(6) $\displaystyle+i(\Delta_{m}+2K|\varpi|^{2}+K).$

In Fig. 2, evolutions of the steady-state microwave-photon numbers and the
magnon number versus the resonant driving power on magnon mode are shown in
two different cases. Figure 2(a) corresponds to case $1$, only cavity $a$ is
driven by the bichromatic microwave fields and cavity $b$ is free of driven.
In case $2$, the same bichromatic microwave fields are only applied on cavity
$b$ in Fig. 2(b). As we can see the steady-state microwave photon numbers and
the magnon number in case $1$ are larger than those in case $2$ even with the
same driving power $P$. In addition, in both cases the microwave-photon
numbers in cavity $a$ are smaller than that in cavity $b$, though the
dissipation of cavity $a$ is lower than that of cavity $b$ and the magnon
mode, i.e., $\kappa_{a}<\kappa_{b},\kappa_{m}$ (on the basis of
$\frac{g}{J}>\frac{1}{3}$ and $K\geq 1n$Hz). The result illustrates that the
counterintuitive asymmetric responds of the steady-state mean values are
affected by the nonlinearity term in equations (2). Meanwhile, it also plays
an important role in evolutions of the perturbation terms.

Expanding equations (2a)-(2c) with the ansatz of operators and removing the
steady-state terms, the perturbation terms can be expressed by the linearized
Heisenberg-Langevin equations

$\displaystyle\delta\dot{a}$ $\displaystyle=$
$\displaystyle-(i\Delta_{c}+\frac{\kappa_{a}}{2})\delta a-iJ\delta
b+\sqrt{\eta_{a}\kappa_{a}}s_{a}e^{-i\Delta_{p}t},$ (7a)
$\displaystyle\delta\dot{b}$ $\displaystyle=$
$\displaystyle-(i\Delta_{c}+\frac{\kappa_{b}}{2})\delta b-iJ\delta a-ig\delta
m$ (7b) $\displaystyle+\sqrt{\eta_{b}\kappa_{b}}s_{b}e^{-i\Delta_{p}t},$
$\displaystyle\delta\dot{m}$ $\displaystyle=$
$\displaystyle-[i(\Delta_{m}+4K|{\varpi}|^{2}+K)+\frac{\kappa_{m}}{2}]\delta
m-ig\delta b$ (7c) $\displaystyle-i2K({\varpi}^{2}\delta
m^{*}+{\varpi}^{*}\delta m^{2}+2{\varpi}\delta m^{*}\delta m)$
$\displaystyle-i2K\delta m^{*}\delta m^{2},$

the quadratic terms $-i2K{\varpi}^{*}\delta m^{2}$, $-i4K{\varpi}\delta
m^{*}\delta m$ in equation (7c) denote the system nonlinearity and can not be
sneezed at. They inevitably cause asymmetric responses of the perturbation
terms, just as the responses of the steady-state mean values in Fig. 2.
However, the cubic term $-i2K\delta m^{*}\delta m^{2}$ in equations (7) is so
small that it will be ignored in the following calculation.

To study asymmetric responses of the perturbations from a new perspective, we
explore the nonreciprocal transmission of the output fields, mainly including
the first-, second- and third-order sidebands, in case $1$ and case $2$. To
get analytical solutions of these high-order sidebands, we define the
perturbation terms have the following forms

$\displaystyle\delta a$ $\displaystyle=$
$\displaystyle{A^{-}_{1}}e^{-i\Delta_{p}t}+{A^{+}_{1}}e^{i\Delta_{p}t}+{A^{-}_{2}}e^{-2i\Delta_{p}t}+{A^{+}_{2}}e^{2i\Delta_{p}t}$
(8a)
$\displaystyle+{A^{-}_{3}}e^{-3i\Delta_{p}t}+{A^{+}_{3}}e^{3i\Delta_{p}t},$
$\displaystyle\delta b$ $\displaystyle=$
$\displaystyle{B^{-}_{1}}e^{-i\Delta_{p}t}+{B^{+}_{1}}e^{i\Delta_{p}t}+{B^{-}_{2}}e^{-2i\Delta_{p}t}+{B^{+}_{2}}e^{2i\Delta_{p}t}$
(8b)
$\displaystyle+{B^{-}_{3}}e^{-3i\Delta_{p}t}+{B^{+}_{3}}e^{3i\Delta_{p}t},$
$\displaystyle\delta m$ $\displaystyle=$
$\displaystyle{M^{-}_{1}}e^{-i\Delta_{p}t}+{M^{+}_{1}}e^{i\Delta_{p}t}+{M^{-}_{2}}e^{-2i\Delta_{p}t}+{M^{+}_{2}}e^{2i\Delta_{p}t}$
(8c)
$\displaystyle+{M^{-}_{3}}e^{-3i\Delta_{p}t}+{M^{+}_{3}}e^{3i\Delta_{p}t},$

this ansatz is rooted from the internal mechanism of the system, where a
series of high-order sidebands Xiong2013  with frequency
$\omega_{d}+n\Delta_{p}$ ($n=1,2,3...$) are generated. $n$ represents the
sideband order as shown in Fig. 1(b). The negative and positive exponents of
the powers are in pairs and respectively denote the upper and lower sidebands.
For examples, the term ${A^{-}_{1}}e^{-i\Delta_{p}t}$ denotes the first-order
upper sideband with coefficient $A^{-}_{1}$, and ${A^{+}_{1}}e^{i\Delta_{p}t}$
corresponds to the first-order lower sideband (the Stokes field Wu2020  in
Fig. 1(b)) with coefficient $A^{+}_{1}$. For the sake of simplicity, in what
follows we only take the high-order upper sidebands into consideration,
because of similar characteristics in the lower sidebands. Of course the
corresponding lower sidebands can also be studied by utilizing the same
method.

With the ansatz we can analytically solve equations (7), the coefficients of
the high-order sidebands can be derived, respectively. Firstly, the
coefficients of the first-order upper sideband are derived as

$\displaystyle B_{1}^{-}$ $\displaystyle=$
$\displaystyle\frac{i\sqrt{\eta_{b}\kappa_{b}}s_{b}(\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})(i\mathcal{C}-\Delta_{p}+2\varpi^{2}K\mathcal{L})}{g^{2}(\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})+(i\mathcal{C}-\Delta_{p}+2{\varpi}^{2}K\mathcal{L})\mathcal{S}_{(-)}}$
$\displaystyle+\frac{iJ\sqrt{\eta_{a}\kappa_{a}}s_{a}(i\mathcal{C}-\Delta_{p}+2\varpi^{2}K\mathcal{L})}{g^{2}(\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})+(i\mathcal{C}-\Delta_{p}+2{\varpi}^{2}K\mathcal{L})\mathcal{S}_{(-)}},$
$\displaystyle A_{1}^{-}$ $\displaystyle=$
$\displaystyle\frac{JB_{1}^{-}+i\sqrt{\eta_{a}\kappa_{a}}s_{a}}{\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2}},$
(9b) $\displaystyle M_{1}^{-}$ $\displaystyle=$
$\displaystyle\frac{-gB_{1}^{-}}{i\mathcal{C}-\Delta_{p}+2K{\varpi}^{2}\mathcal{L}},$
(9c) $\displaystyle M_{1}^{+}$ $\displaystyle=$
$\displaystyle\mathcal{L}^{*}M_{1}^{-*},$ (9d)

with

$\displaystyle\mathcal{C}$ $\displaystyle=$
$\displaystyle-i(\Delta_{m}+K+4K|{\varpi}|^{2})-\frac{\kappa_{m}}{2},$ (10)
$\displaystyle\mathcal{S}_{(\pm)}$ $\displaystyle=$
$\displaystyle(\Delta_{p}\pm\Delta_{c}+i\frac{\kappa_{a}}{2})(\Delta_{p}\pm\Delta_{c}+i\frac{\kappa_{b}}{2})-J^{2},$
(11) $\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{2K\mathcal{S}_{(+)}{\varpi}^{2*}}{g^{2}(\Delta_{p}+\Delta_{c}+i\frac{\kappa_{a}}{2})-(\Delta_{p}-i\mathcal{C})\mathcal{S}_{(+)}}.$
(12)

For the second-order upper sideband, the coefficients are given as

$\displaystyle B_{2}^{-}$ $\displaystyle=$
$\displaystyle\frac{g(2\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})M_{2}^{-}}{G},$
(13a) $\displaystyle A_{2}^{-}$ $\displaystyle=$
$\displaystyle\frac{JB_{2}^{-}}{2\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2}},$
(13b) $\displaystyle M_{2}^{-}$ $\displaystyle=$
$\displaystyle\frac{-2KG(\varpi^{2}E^{*}+\varpi^{*}M_{1}^{-2}+2\varpi
M_{1}^{-}M_{1}^{+*})}{(i\mathcal{C}-2\Delta_{p}+2K\varpi^{2}D^{*})G+g^{2}(2\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})},$
$\displaystyle M_{2}^{+}$ $\displaystyle=$ $\displaystyle DM_{2}^{-*}+E,$
(13d)

where

$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{2\varpi^{2}KF}{g^{2}(2\Delta_{p}+\Delta_{c}-i\frac{\kappa_{a}}{2})-F(2\Delta_{p}+i\mathcal{C})},$
(14) $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{2\varpi^{*}KFM_{1}^{+2}+4\varpi
KFM_{1}^{+}M_{1}^{-*}}{g^{2}(2\Delta_{p}+\Delta_{c}-i\frac{\kappa_{a}}{2})-F(2\Delta_{p}+i\mathcal{C})},$
(15) $\displaystyle F$ $\displaystyle=$
$\displaystyle(2\Delta_{p}+\Delta_{c}-i\frac{\kappa_{b}}{2})(2\Delta_{p}+\Delta_{c}-i\frac{\kappa_{a}}{2})-J^{2},$
(16) $\displaystyle G$ $\displaystyle=$
$\displaystyle(2\Delta_{p}-\Delta_{c}+i\frac{\kappa_{b}}{2})(2\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})-J^{2}.$
(17)

To the third-order upper sideband, the coefficients have the following forms

$\displaystyle B_{3}^{-}$ $\displaystyle=$
$\displaystyle\frac{g(3\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})M_{3}^{-}}{(3\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})(3\Delta_{p}-\Delta_{c}+i\frac{\kappa_{b}}{2})-J^{2}},$
(18a) $\displaystyle A_{3}^{-}$ $\displaystyle=$
$\displaystyle\frac{JB_{3}^{-}}{3\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2}},$
(18b) $\displaystyle M_{3}^{-}$ $\displaystyle=$
$\displaystyle\frac{2K\varpi^{2}H_{+}^{*}+H_{-}(3\Delta_{c}+gT^{*}-i\mathcal{C}^{*})}{(3i\Delta_{p}+\mathcal{C}-igQ)(-3\Delta_{p}-gT^{*}+i\mathcal{C}^{*})-4iK^{2}|\varpi|^{4}},$
$\displaystyle M_{3}^{+}$ $\displaystyle=$
$\displaystyle\frac{2\varpi^{2}KM_{3}^{-*}+iH_{+}}{-3\Delta_{p}-gB_{3}-i\mathcal{C}},$
(18d)

in which

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{g(3\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})}{(3\Delta_{p}-\Delta_{c}+i\frac{\kappa_{a}}{2})(3\Delta_{p}-\Delta_{c}+i\frac{\kappa_{b}}{2})-J^{2}},$
(19) $\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{-g(3\Delta_{p}+\Delta_{c}-i\frac{\kappa_{a}}{2})}{(3\Delta_{p}+\Delta_{c}-i\frac{\kappa_{a}}{2})(3\Delta_{p}+\Delta_{c}-i\frac{\kappa_{b}}{2})-J^{2}},$
(20) $\displaystyle H_{+}$ $\displaystyle=$
$\displaystyle-4i\varpi^{*}KM_{1}^{+}M_{2}^{+}-4i\varpi
K(M_{1}^{+}M_{2}^{-*}+M_{2}^{+}M_{1}^{-*}),$ $\displaystyle H_{-}$
$\displaystyle=$ $\displaystyle-4i\varpi^{*}KM_{1}^{-}M_{2}^{-}-4i\varpi
K(M_{2}^{-}M_{1}^{+*}+M_{1}^{-}M_{2}^{+*}).$

It can be seen that the coefficients $A_{1}^{-}$, $B_{1}^{-}$, $A_{2}^{-}$,
$B_{2}^{-}$, $A_{3}^{-}$ and $B_{3}^{-}$ of the high-order sidebands are
closely dependent on the magnon mode including the steady-state amplitude and
the perturbation terms.

With above analytical solutions, we calculate the output fields of each
sideband in two cases. In case $1$, the control and probe fields are injected
from cavity $a$ and the output fields from cavity $b$ are explored. With
input-output relation, the output fields of high-order sidebands from cavity
$b$ can be expressed as

$\displaystyle s_{{aout}}$ $\displaystyle=$
$\displaystyle\sqrt{\eta_{b}\kappa_{b}}B_{1}^{-}e^{-i(\omega_{d}+\Delta_{p})t}+\sqrt{\eta_{b}\kappa_{b}}B_{2}^{-}e^{-i(\omega_{d}+2\Delta_{p})t}$
(23)
$\displaystyle+\sqrt{\eta_{b}\kappa_{b}}B_{3}^{-}e^{-i(\omega_{d}+3\Delta_{p})t}.$

In the direction from cavity $a$ to $b$, the transmission coefficients of the
first-, second- and third-order sidebands are respectively given as

$\displaystyle t_{ba}$ $\displaystyle=$
$\displaystyle\left|\frac{\sqrt{\eta_{b}\kappa_{b}}B_{1}^{-}}{s_{a}}\right|,$
(24a) $\displaystyle\tau_{ba}$ $\displaystyle=$
$\displaystyle\left|\frac{\sqrt{\eta_{b}\kappa_{b}}B_{2}^{-}}{s_{a}}\right|,$
(24b) $\displaystyle\ell_{ba}$ $\displaystyle=$
$\displaystyle\left|\frac{\sqrt{\eta_{b}\kappa_{b}}B_{3}^{-}}{s_{a}}\right|.$
(24c)

In case $2$, the same control and probe fields are applied on cavity $b$ and
the output fields from cavity $a$ are researched. In order to avoid confusion,
we use $\mathcal{A}^{-}_{j}$, $\mathcal{B}^{-}_{j}$ and $\mathcal{M}^{-}_{j}$
($j=1,2,3$) to distinguish from the coefficients in case $1$. Combining this
condition and the input-output relation, output fields of the high-order upper
sidebands are given as follows

$\displaystyle s_{{bout}}$ $\displaystyle=$
$\displaystyle\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}_{1}^{-}e^{-i(\omega_{d}+\Delta_{p})t}+\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}_{2}^{-}e^{-i(\omega_{d}+2\Delta_{p})t}$
(25)
$\displaystyle+\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}_{3}^{-}e^{-i(\omega_{d}+3\Delta_{p})t}.$

Similarly, along the direction from cavity $b$ to $a$, the transmission
coefficients of the first-, second- and third-order sidebands are defined as

$\displaystyle t_{ab}$ $\displaystyle=$
$\displaystyle\left|\frac{\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}_{1}^{-}}{s_{b}}\right|,$
(26a) $\displaystyle\tau_{ab}$ $\displaystyle=$
$\displaystyle\left|\frac{\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}_{2}^{-}}{s_{b}}\right|,$
(26b) $\displaystyle\ell_{ab}$ $\displaystyle=$
$\displaystyle\left|\frac{\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}_{3}^{-}}{s_{b}}\right|.$
(26c)

In order to more clearly describe nonreciprocal transmissions of the high-
order sidebands in two cases, we introduce definition of the nonreciprocal
isolation ratio, which is expressed as

$\displaystyle I_{o}=\left|{\rm
lo}g_{10}\frac{|\sqrt{\eta_{b}\kappa_{b}}B^{-}_{j}|^{2}}{|\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}^{-}_{j}|^{2}}\right|,$
(27)

here $o$ stands for either $t$, $\tau$ or $\ell$ corresponding to $j=1,2,3$.
$I_{t}$, $I_{\tau}$ and $I_{\ell}$ respectively denote isolation ratio of the
first-, second- and third-order sidebands. Then we use isolation ratio $I_{o}$
to quantitatively describe degree of the nonreciprocal transmissions in two
cases. It is decided by the value of $I_{o}$: if $I_{o}=0$, equivalently,
$|\sqrt{\eta_{b}\kappa_{b}}B^{-}_{j}|^{2}/|\sqrt{\eta_{a}\kappa_{a}}\mathcal{A}^{-}_{j}|^{2}=1$,
it means transmissions of the high-order sidebands are reciprocal (symmetry)
along two opposite directions. For other situations, the higher the value of
$I_{o}$, the stronger is the nonreciprocity of the transmissions in two cases.
For example, $I_{o}\geq 1$, which illustrates transmission coefficient in one
direction is at least $10$ times that of the other direction. It also shows
that transmittances along two opposite directions have a strong asymmetry.
This situation can be approximated as one-way transmission of high-order
sidebands.

##  III Nonreciprocal transmissions of the high-order sidebands in a cavity
magnonics system

Since the transmission coefficients in equations (24) and (26) and the
isolation ratio in equation (27) are mainly dominated by coefficients of each
sideband, which are closely associated with the steady-state amplitude and the
perturbation terms of magnon mode. Under this situation, we analyze influence
of the driving power, applied on the magnon mode, and the frequency detuning
between the driving field and the magnon mode on the nonreciprocal
transmission of each sideband.

Figure 3: (Color online) The logarithmic amplitudes ${\rm lo}g_{10}{\rm
S}(\omega)$ of the transmitted high-order sidebands vs frequency
$\omega/\Delta_{p}$ in case $1$ and case $2$. Here $P=15\,\,m$W,
$\Delta_{p}/\Delta_{c}=0.7$, other parameters are same as in Fig. 2.

For the same purpose, but utilizing a different method, we first exhibit an
overview comparison of the high-order sideband spectrums between case $1$ and
case $2$. Here the magnon mode is resonantly driven by a $15\,\,m$W driving
field. As shown in Fig. 3, spectrums of the adjacent sidebands are equally
spaced apart from each other at frequency $\Delta_{p}$ in the rotating frame.
Where $\omega/\Delta_{p}=0$ denotes the control field, $\omega/\Delta_{p}=n$
($n=1,2,3$) corresponds to the nth-order sideband. It can be seen that there
exists amplitude gap between the sidebands with the same order in case $1$ and
case $2$. Scilicet, the transmissions of the high-order sidebands are
nonreciprocal in two cases. What’s more, amplitudes of the high-order
sidebands decrease rapidly as the sideband order increases. Specifically,
$S(\Delta_{p})/S(0)<10^{-3}$, $S(2\Delta_{p})/S(0)<10^{-7}$ and
$S(3\Delta_{p})/S(0)<10^{-12}$ in both cases. This proves that it is valid to
regard high-order sidebands as perturbation compared with the strong control
field. Then, we analyze in detail the transmission nonreciprocity of each
sideband in the following content.

###  III.1 Dependence of nonreciprocal transmissions for high-order sidebands
on the resonant driving power $P$

Figure 4 shows transmission coefficients $|t_{ba}|^{2}$ and $|t_{ab}|^{2}$
vary with frequency detuning $\delta/\Delta_{c}$ under different resonant
driving powers $P$. In Fig. 4(a), we choose driving power $P=0$. Under this
situation, $|t_{ba}|^{2}$ and $|t_{ab}|^{2}$ are smaller than $0.2$ and
evolutions of them are synchronous.

Figure 4: (Color online) Transmission coefficients $|t_{ba}|^{2}$ and
$|t_{ab}|^{2}$ of the first-order sideband vs frequency detuning
$\delta/\Delta_{c}$. To observe transmission coefficients clearly,
$|t_{ab}|^{2}$ in (b) and $|t_{ba}|^{2}$ in (c) have been increased by one
order of magnitude. The driving powers $P=0$, $20\,\,m$W, $30\,\,m$W in (a),
(b) and (c), respectively. Other parameters are same as that in Fig. 2.

It means the transmissions of the first-order sideband in two cases are
reciprocal when the driving field on magnon mode is absent. Increasing driving
power to $20\,\,m$W in Fig. 4(b), intuitively, only a peak with value $150$ of
$|t_{ba}|^{2}$ is seen at $\delta/\Delta_{c}=-0.3$; and a peak value $5.6$ of
$|t_{ab}|^{2}$ is shown at $\delta/\Delta_{c}=-0.25$. However, if one zooms in
locally, a local peak of $|t_{ba}|^{2}$ and a tiny one of $|t_{ab}|^{2}$ can
be found at the position of the blue-dotted line. This illustrates that the
first-order sideband (probe field) is magnified hundreds of times compared
with the original input probe field and mainly transmitted at
$\delta/\Delta_{c}=-0.3$ in case $1$. Whereas, in case $2$, magnification and
transmission of the first-order sideband mainly occurs at
$\delta/\Delta_{c}=-0.25$. From another point of view, the transmissions of
the first-order sideband in two cases are robustly nonreciprocal near
$\delta/\Delta_{c}=-0.3$. Continue to enhance the driving power to $30\,\,m$W
in Fig. 4(c), we find $|t_{ba}|^{2}$ with two peaks is always smaller than
$2$; while $|t_{ab}|^{2}$ reaches its maximum value $15.5$ at
$\delta/\Delta_{c}=-0.31$, with the other invisible peak at the blue-dotted
line. The results illustrate that the first-order sideband is magnified less
than twice in case $1$, but more than ten times in case $2$; and the strong
transmission nonreciprocity exists near $\delta/\Delta_{c}=-0.3$. From above
numerical analysis, we find the transmission nonreciprocity of the first-order
sideband is effectively modulated by the driving power acted on the magnon
mode. This stems from the fact that the increased driving power excites more
magnon polarons

Figure 5: (Color online) Transmission coefficients $\tau_{ba}$, $\tau_{ab}$
and $\ell_{ba}$, $\ell_{ab}$ of the second- and third-order sidebands vs
frequency detuning $\delta/\Delta_{c}$. In order to clearly observe the
transmission coefficients, the values of $\tau_{ab}$ in (b) and $\ell_{ab}$ in
(e) have been respectively magnified by three and five orders of magnitude.
Here $P=0$ in panels (a) and (d), $P=20\,\,m$W in panels (b) and (e), and
$P=30\,\,m$W in panels (c) and (f). Other parameters are same as that in Fig.
2.

and simultaneously enhances the effective Kerr nonlinearity of the magnon,
which eventually strengthens the nonlinear asymmetric responses of the system
in two cases. In addition, it needs to be emphasized that peaks on each curve
should have been resonant with the effective supermodes made up of two
microwave cavity modes. Frequencies of the supermodes shift
$\pm\sqrt{J^{2}-(\frac{\kappa_{b}-\kappa_{a}}{2})^{2}}$ (equal to $\pm
0.3\Delta_{c}$) from the original cavity frequency $\omega_{c}$. This is
originated from the strong cavity-cavity coupling rate, i.e.,
$J>\kappa_{a},\kappa_{b}$ in the system. Practically, the actural resonant
frequency exhibit shifts due to the joint interference of the strong cavity-
magnon coherent interaction ($g>J$) and the magnon Kerr nonlinearity Kons2015
.

Figure 5 shows transmission coefficients $\tau_{ba}$, $\tau_{ab}$ and
$\ell_{ba}$, $\ell_{ab}$ of the second- and the third-order sidebands versus
frequency detuning $\delta/\Delta_{c}$ with different resonant driving powers.
When the driving field is absent in Figs. 5(a) and 5(d), although the
transmission coefficients are very tiny, the transmission nonreciprocity is
very obvious in the regime $\delta/\Delta_{c}\in[-0.5\,\,-0.2]$. Once the
driving field is considered, both the transmission coefficients and
nonreciprocity will be further enhanced. For instance, $P=20\,\,m$W in Figs.
5(b) and 5(e), $\tau_{ba}$ and $\ell_{ba}$ reach respective unique maximum
$0.7$ and $1.5\times 10^{-3}$ at $\delta/\Delta_{c}=-0.3$. While $\tau_{ab}$
and $\ell_{ab}$ keep much smaller maximums. Continue to increase the driving
power to $30\,\,m$W in Fig. 5(c) and 5(f). Both $\tau_{ba}$ and $\ell_{ba}$
have very tiny maximums. While $\tau_{ab}$ and $\ell_{ab}$ have soared to
their peaks $4.8\times 10^{-3}$ and $4.6\times 10^{-7}$ at the point
$\delta/\Delta_{c}=-0.31$. This illustrates that the resonant driving field
can not only effectively manipulate the transmission nonreciprocity of the
first-order sideband, but also of the second- and the third-order sidebands.
What’s more, the transmission nonreciprocity of the high-order sidebands have
a cooperative and consistent characteristic when facing different driving
powers. Besides being applied in high performance nonreciprocal directional-
switching

Figure 6: (Color online) The isolation ratio $I_{t}$ of the first-order
sideband vs the resonant driving power $P$ and frequency detuning
$\delta/\Delta_{c}$. Parameters are the same as that in Fig. 2.

isolator Assanto2008 ; Khanikaev2020  and diode Gong2011 ; Zhu2013 ; Qi2012 ,
such a series of equally spaced high-order sidebands also have potential
applications in frequency comb-like precision measurement through simply
operating the driving power.

Then, we quantitatively describe the degree of transmission nonreciprocity for
each sideband by isolation ratio. Figure 6 intuitively displays isolation
ratio $I_{t}$ varies with the resonant driving power $P$ and frequency
detuning $\delta/\Delta_{c}$. As the driving power increases, one bright
region, composed of an upper and a lower parts, appears at the center of
$\delta/\Delta_{c}=-0.3$. In this region the optimal $I_{t}$ is $5.6$ dB
obtained with $P=21.8\,\,m$W. This can also be explained by the result in Fig.
4, i.e., the prominent large differences of transmission coefficients
$|t_{ba}|^{2}$ and $|t_{ab}|^{2}$ between the left resonant peaks near
$\delta/\Delta_{c}=-0.3$.

For the second- and third-order sidebands, the corresponding isolation ratios
$I_{\tau}$ and $I_{\ell}$ versus the driving power $P$ and frequency detuning
$\delta/\Delta_{c}$ are given in Fig. 7(a) and 7(b). In Fig. 7(a), the value
of $I_{\tau}$ in most areas is lower than $3.5$ dB, except two bright areas
centered at $\delta/\Delta_{c}=-0.3$, where $I_{\tau}$ reaches its maximum
$9.9$ dB. While the distribution of $I_{\ell}$ is quite different. As
presented in Fig. 7(b), except a small dark area, the values of $I_{\ell}$ in
other areas are higher than $10$ dB. Especially in the yellow area centered at
$\delta/\Delta_{c}=-0.4$ with $P\in[0\,\,10]\,m$W and the other one centered
at $\delta/\Delta_{c}=-0.3$ with $P\in[22\,\,28]\,m$W, $I_{\ell}$ can be
higher than $30$ dB. By an overall observing, what $I_{\ell}$ and $I_{\tau}$
have in common is that the optimal value is distributed near
$\delta/\Delta_{c}=-0.3$, this is consistent with the result in Fig. 5.

Taken together Fig. 6 and Fig. 7, we find that the isolation ratios $I_{t}$,
$I_{\tau}$ and $I_{\ell}$ show an overall increasing trend with the increase
of the sideband order. Specifically, optimal $I_{t}$, $I_{\tau}$ and
$I_{\ell}$ are enhanced in turn from $5.6$ dB, to $9.9$ dB and lastly $30$ dB
near detuning $\delta/\Delta_{c}=-0.3$. The reason is rooted from the
strengthened effective Kerr nonlinearity in each sideband. Besides the steady-
state amplitude of the magnon mode, the first-order sideband is also
associated with the first-order perturbation terms (in equations (9)) of the
magnon mode; the second-order sideband is simultaneously related to the first-
and the

Figure 7: (Color online) isolation ratios (a) $I_{\tau}$ of the second-order
sideband and (b) $I_{\ell}$ of the third-order sideband vs the resonant
driving power $P$ and frequency detuning $\delta/\Delta_{c}$. Parameters are
same as that in Fig. 2.

second-order perturbation terms of the magnon mode (in equations. (13)); and
the third-order sideband is totally related to the first-, second- and third-
order perturbation terms (in equations (18)) of the magnon mode. Except for
the steady-state amplitude of the magnon, the strong transmission
nonreciprocity of each sideband originates from the aggregate nonlinear
effects in each perturbation term. Therefor, the higher the order of the
sideband, the stronger nonlinear asymmetry of the transmission, it lastly
leads to much higher isolation ratio.

Figure 8: (Color online) The isolation ratios (a) $I_{t}$, (b) $I_{\tau}$ and
(c) $I_{\ell}$ vs $\delta/\Delta_{c}$ with different detunings $\Delta_{m}$
between the magnon mode and the correspong driving field. Here $P=25\,\,m$W,
and the other parameters are the same as in Fig. 2.

###  III.2 Dependence of the nonreciprocal transmissions for high-order
sidebands on frequency detuning $\Delta_{m}$

In this section, we research influence of frequency detuning $\Delta_{m}$ on
the nonreciprocal transmission of the high-order sidebands. In Fig. 8,
Isolation ratios $I_{t},~{}I_{\tau}$ and $I_{\ell}$ vs $\delta/\Delta_{c}$
with different detunings $\Delta_{m}$ are intuitively given. When magnon mode
is resonant with the driving field i.e., $\Delta_{m}=0$, isolation ratios
$I_{t},~{}I_{\tau}$ and $I_{\ell}$ increase in turn. Especially in the optimal
detuning regime (centered at the resonant frequency of one effective supermode
of the microwave cavities Kons2015 ), $I_{t}$, $I_{\tau}$ and $I_{\ell}$ reach
their maximum values. The same evolution trend is also suitable for non-
resonant situations, i.e., $\Delta_{m}/2\pi=-19.2$ MHz, $20$ MHz, except for
the reduced optimal isolation ratios and the changed optimal detuning regimes.
It is because excitation of the magnetic polarons is suppressed in the non-
resonant situations and results in much weaker effective Kerr nonlinearity.
This lastly induces much weaker transmission nonreciprocity and shifts the
optimal detuning regime.

For actual operation to obtain strong nonreciprocity of the high-order
sidebands, the bandwidth of the optimal detuning regime is often more
concerned. In the resonant situation, we centralize $I_{t}$, $I_{\eta}$ and
$I_{\ell}$ in Fig. 9. It can be obviously seen $I_{t}$, $I_{\eta}$ and
$I_{\ell}$

Figure 9: (Color online) The isolation ratios $I_{t}$, $I_{\tau}$ and (c)
$I_{\ell}$ vs $\delta/\Delta_{c}$ when the magnon mode is resonant with the
driving field, i.e., $\Delta_{m}=0$. Here $P=25\,\,m$W and the other
parameters are the same as in Fig. 2.

increase in turn and reach their optimal values in the gray shaded area. This
means an overall controlling of the strong nonreciprocity for high-order
sidebands can be realized in the optimal detuning regime ranging from
$\delta/\Delta_{c}=-0.301$ to $\delta/\Delta_{c}=-0.273$. The bandwidth of the
optimal detuning regime is seven megahertz with $\Delta_{c}=2\pi\times 40$ MHz
in the system. It is achievable for experimental operation to capture strong
transmission nonreciprocity of the high-order sidebands simultaneously in such
a bandwidth.

##  IV Conclusion

In summary, we have theoretically provided a method to realize an overall
controlling of the strong transmission nonreciprocity of the first-, second-
and third-order sidebands in a cavity magnonics system. It is consist of two
coupled microwave cavities and one YIG (or another suitable magnonic material)
sphere. Our approach utilizes the self-Kerr nonlinearity of magnon in the YIG
sphere. We have shown that strong transmission nonreciprocity of the high-
order sidebands can be achieved by respectively regulating the driving power
on the magnon mode and the frequency detuning between the magnon mode and the
driving field. We also showed that the higher the order of the sideband, the
stronger is the nonreciprocity marked by the enhanced isolation ratio in the
optimal detuning regime. We have illustrated that the bandwidth of the optimal
detuning regime can reach several megahertz when magnon mode is resonant
drived. This implies that it is experimentally feasible to capture robust
transmission nonreciprocity of the high-order sidebands simultaneously. This
study provides a promising route to realize strong nonreciprocity of the high-
order sidebands at microwave frequencies and has potential applications in
frequency comb-like precision measurement.

###### Acknowledgements.

We acknowledge professor Xin-You Lü for his valuable suggestions on our work.
And this work was supported by the National Natural Science Foundation of
China (NSFC) under Grants 12005078, 11675124 and 11704295.

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