# Kerr Nonlinearity Induced Nonreciprocity in dissipatively coupled resonators

Qingtian Miao<EMAIL_ADDRESS>Department of Physics and Astronomy, Texas A&M
University, Texas 77843, USA  G.S. Agarwal<EMAIL_ADDRESS>Institute for Quantum Science and Engineering, Department of Biological and
Agricultural Engineering, Department of Physics and Astronomy, Texas A&M
University, College Station, Texas 77843, USA

###### Abstract

Nonlinearity induced nonreciprocity is studied in a system comprising two
resonators coupled to a one-dimensional waveguide when the linear system does
not exhibit nonreciprocity. The analysis is based on the Hamiltonian of the
coupled system and includes the dissipative coupling between the waveguide and
resonators, along with the input-output relations. We consider a large number
of scenarios which can lead to nonreciprocity. We pay special attention to the
case when the linear system does not exhibit nonreciprocal behavior. In this
case, we show how very significant nonreciprocal behavior can result from Kerr
nonlinearities. We find that the bistability of the nonlinear system can aid
in achieving large nonreciprocity. Additionally, We bring out nonreciprocity
in the excitation of each resonator, which can be monitored independently. Our
results highlight the profound influence of nonlinearity on nonreciprocal
behavior, offering a new avenue for controlling light propagation in
integrated photonic circuits. Nonlinearity induced nonreciprocity would lead
to significant nonreciprocity in quantum fluctuations when our system is
treated quantum mechanically.

##  I Introduction

The nonreciprocal propagation of light is increasingly attracting attention
[1, 2, 3, 4, 5]. Time reversal symmetry breaking is a necessary condition for
nonreciprocity. Historically, this has been achieved with magneto-optical
(Faraday-rotation) crystals that require an external magnetic bias. However,
these crystals are not compatible with semiconductor chip integration due to
their bulkiness and complexity. More generally, nonreciprocity often naturally
arises if the medium is chiral as is typical in magnetic systems [6]. Recent
studies in chiral quantum optics have shown that strong light confinement in
certain structures can lock the local light polarization along its direction
of travel, yielding direction-dependent emission characteristics [7]. It is
more challenging to produce nonreciprocity in achiral systems. Some standard
demonstrations of nonreciprocity use phase matching conditions in a nonlinear
medium [8, 9]. Clearly, the phase matching conditions can not be
simultaneously satisfied for two opposite directions of propagation and this
would result in nonreciprocity. A recent demonstration of nonreciprocity is
based on the intrinsic nonlinearities of two-level superconducting artificial
atoms [10]. Additionally, the use of synthetic electric and magnetic fields,
or a suitable modulation of the refractive index, are attracting considerable
attention for producing nonreciprocity without the need for traditional
magnetic fields [11, 12, 13, 14, 15, 16, 17, 18]. Synthetic fields are
preferable over magnetic fields [19, 20] as implementation of synthetic fields
is straightforward. It should be noted that the nonreciprocal transport has
consequences in several other contexts like heat transport [21, 22, 23, 24],
one-way amplification [25]; quantum fluctuations and quantum entanglement [26,
27, 28, 29, 30]. One would like to have systems that are scalable and
integrable leading to the theoretical development and experimental realization
of nonreciprocal devices such as isolators, circulators, and directional
amplifiers.

Waveguide-coupled resonator systems are increasingly capturing the interest of
researchers for their pivotal role in facilitating nonreciprocal light
propagation, a critical mechanism for unidirectional wave transport, and a
foundational element for future quantum-information networks. These systems
are especially attractive because of the waveguide-mediated dissipative
coupling and the resulting bound states in continuum which can be instrumental
in enhancing nonreciprocal effects [10, 25, 31, 32, 33]. One way to produce
nonreciprocity in such systems is to use both dissipative and dispersive
couplings among resonators. Unidirectional propagation was demonstrated using
the combined effect of the dispersive and dissipative couplings [6]. The
question that we address in this work is how to achieve nonreciprocity if the
resonators are far apart so that dispersive coupling is almost zero. One way
is to use Kerr nonlinearity and unidirectional propagation of waves inside the
resonators as would be the case of resonators coupled by a waveguide [34, 35,
36]. This is different from the bidirectional propagation inside the
resonator, which always produces nonreciprocity as demonstrated in several
experiments [37, 38]. Generally, Kerr nonlinearity has been small for most
materials and requires higher powers however with the development of
superconducting circuits, this is no longer the case as the Kerr nonlinearity
is then fairly large [39].

The manuscript is organized as follows. In Section II we present details of
our model and use semiclassical approach and temporal coupled-mode theory
(TCMT) [40, 41] to obtain basic equations for the fields in the two resonators
dissipatively coupled to a waveguide. We also present input-output relations.
In Section III we derive equations for the transmission amplitudes which
depend nonlinearly on the fields in the nonlinear medium. In Section IV, we
derive conditions when nonreciprocity without nonlinearity is possible and
establish connection with some of the existing results. In Section V, we
conduct a thorough analysis to understand how the system’s parameters affect
nonreciprocal behavior, highlighting the substantial nonreciprocity due to
nonlinearity. In certain parameter domain the system exhibits bistability and
then we present nonreciprocity in bistable transmission. In Section VI we give
nonreciprocity in the excitation of the resonators. In Section VII we discuss
the popular system consisting of a magnetic system coupled dissipatively to a
resonator [42, 6, 43]. Our insights not only enhance the comprehension of
nonreciprocal behavior in both linear and nonlinear systems but also
facilitate the development of novel integrated photonic circuit designs, where
the control of light propagation is crucial. We finally conclude with remarks
about nonreciprocity in quantum fluctuations if the system is described by
full quantum theory.

##  II Model and Basic Equations

Figure 1: Two resonators $a$ and $b$, coupled through a parameter $J$, are
connected to a one-dimensional waveguide, facilitating the interactions with
incoming and outgoing waves. The separation $x_{ab}$ between resonators leads
to a phase shift in wave propagation. $\Gamma_{a,b}$ denotes the external
damping rates, and $\gamma_{a,b}$ denotes the intrinsic damping rates.

We consider a system composed of two resonators $a$ and $b$ coupled to a one-
dimensional waveguide, as depicted in Fig. 1. The resonators have resonance
frequencies $\omega_{a}$ and $\omega_{b}$, respectively. Resonator $a$ can
exhibit Kerr nonlinearity. Positioned at a distance $x_{ab}$ apart, the two
resonators are coupled directly via a complex parameter $J$. In addition, they
also interact through the propagating waves in the waveguide. The waveguide
modes, both incoming ($\varepsilon_{1\rightarrow}$,
$\varepsilon_{2\leftarrow}$) and outgoing ($\varepsilon_{1\leftarrow}$,
$\varepsilon_{2\rightarrow}$), where $1,2$ specifies the port, are coherent
and operating at the driving frequency $\omega_{d}$. The resonant modes are
excited by the incoming waves ($\varepsilon_{1\rightarrow}$,
$\varepsilon_{2\leftarrow}$) of two ports with some coupling constants
$k_{a\ell}$, $k_{b\ell}$ ($\ell=1,2$). We can write the Hamiltonian of the
system (we set $\hbar=1$) in the rotating frame of drive field as

$\displaystyle H$
$\displaystyle=\Delta_{a}a^{\dagger}a+\Delta_{b}b^{\dagger}b+Ja^{\dagger}b+J^{*}b^{\dagger}a+Ua^{\dagger
2}a^{2}$ (1)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+i(k_{a1}\varepsilon_{1\rightarrow}a^{\dagger}-k_{a1}^{*}a\varepsilon_{1\rightarrow}^{\dagger})+i(k_{a2}e^{i\phi}\varepsilon_{2\leftarrow}a^{\dagger}-k_{a2}^{*}e^{-i\phi}a\varepsilon_{2\leftarrow}^{\dagger})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+i(k_{b1}e^{i\phi}\varepsilon_{1\rightarrow}b^{\dagger}-k_{b1}^{*}e^{-i\phi}b\varepsilon_{1\rightarrow}^{\dagger})+i(k_{b2}\varepsilon_{2\leftarrow}b^{\dagger}-k_{b2}^{*}b\varepsilon_{2\leftarrow}^{\dagger}).$

Here, $\Delta_{a}=\omega_{a}-\omega_{d}$, $\Delta_{b}=\omega_{b}-\omega_{d}$
are the detuning of the resonant frequencies $\omega_{a}$ and $\omega_{b}$
compared to the driving frequency, $\omega_{d}$. $U$ quantifies the strength
of third-order nonlinearity in mode $a$. The phase shift is
$\phi=\omega_{d}x_{ab}/v_{p}$, experienced by the drive as it travels between
the resonators, with $v_{p}$ denoting the phase velocity of the drive.

In what follows, we use the semiclassical description and adopt the TCMT. The
dynamics of the system in the rotating frame of the drive can be given as

$\displaystyle\frac{d}{dt}\mathbf{c}$
$\displaystyle=-i\mathbf{\Delta}\mathbf{c}-\mathbf{\Gamma}\mathbf{c}+\mathbf{K}\pmb{\varepsilon}_{\rm
in}$ (2) $\displaystyle\\!\\!\\!\\!\\!\\!-\left(\begin{array}[]{cc}0&iJ\\\
iJ^{*}&0\end{array}\right)\mathbf{c}-2iU(a^{\dagger}a)\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\mathbf{c},$ $\pmb{\varepsilon}_{\rm
out}=\mathbf{C}\left(\pmb{\varepsilon}_{\rm
in}-\mathbf{D}^{\dagger}\mathbf{c}\right),$ (3)

where $\mathbf{c}=(a,b)^{T}$, $\pmb{\varepsilon}_{\rm
in}=(\varepsilon_{1\rightarrow},\varepsilon_{2\leftarrow})^{T}$,
$\pmb{\varepsilon}_{\rm
out}=(\varepsilon_{1\leftarrow},\varepsilon_{2\rightarrow})^{T}$,
$\mathbf{\Delta}=\left(\begin{array}[]{cc}\Delta_{a}-i\gamma_{a}&0\\\
0&\Delta_{b}-i\gamma_{b}\end{array}\right)$ with the intrinsic damping rates
$\gamma_{a}$, $\gamma_{b}$ due to the material loss included,
$\mathbf{C}=\left(\begin{array}[]{cc}0&e^{i\phi}\\\
e^{i\phi}&0\end{array}\right)$ for the direct scattering in the absence of
resonators. The matrix $\mathbf{\Gamma}$ characterizes the exponential decay
process. The matrices $\mathbf{K}$ and $\mathbf{D}^{\dagger}$ describe the
coupling of the resonator to the input and output fields, respectively. The
coupling matrix $\mathbf{K}$ are followed based on the structure of the
Hamiltonian [Eq. (1)],

$\mathbf{K}=\left(\begin{array}[]{cc}k_{a1}&k_{a2}e^{i\phi}\\\
k_{b1}e^{i\phi}&k_{b2}\end{array}\right).$ (4)

In order to connect the $\mathbf{\Gamma}$ and $\mathbf{D}$ matrices to the
coupling parameters $k_{a\ell}$ and $k_{b\ell}$, we look into the energy
conservation constraint when $\gamma_{a}=\gamma_{b}=J=U=0$. Followed by energy
conservation, $d(\mathbf{c}^{\dagger}\mathbf{c})/dt=\pmb{\varepsilon}_{\rm
in}^{\dagger}\pmb{\varepsilon}_{\rm in}-\pmb{\varepsilon}_{\rm
out}^{\dagger}\pmb{\varepsilon}_{\rm out}$. With an assumption of no incoming
waves, $\pmb{\varepsilon}_{\rm in}=0$, the rate of change of energy for
resonators becomes
$d(\mathbf{c}^{\dagger}\mathbf{c})/dt=d(a^{\dagger}a+b^{\dagger}b)/dt=-\mathbf{c}^{\dagger}(\mathbf{\Gamma}+\mathbf{\Gamma}^{\dagger})\mathbf{c}$,
and the outgoing power is $\pmb{\varepsilon}_{\rm
out}^{\dagger}\pmb{\varepsilon}_{\rm
out}=\mathbf{c}^{\dagger}\mathbf{D}\mathbf{D}^{\dagger}\mathbf{c}$. The energy
conservation constraint in this case leads to
$\mathbf{D}\mathbf{D}^{\dagger}=\mathbf{\Gamma}+\mathbf{\Gamma}^{\dagger}$.
For $\pmb{\varepsilon}_{\rm in}\neq 0$ and at steady state where
$d\mathbf{c}/dt=0$, we find
$\mathbf{D}\mathbf{K}^{-1}\mathbf{\Delta}=\mathbf{\Delta}\mathbf{K}^{-1\dagger}\mathbf{D}^{\dagger}$
and
$\mathbf{\Gamma}^{\dagger}\mathbf{K}^{-1\dagger}\mathbf{D}^{\dagger}+\mathbf{D}\mathbf{K}^{-1}\mathbf{\Gamma}=\mathbf{\Gamma}+\mathbf{\Gamma}^{\dagger}$,
valid for any matrix $\mathbf{\Delta}$ [40, 41]. Thus we can determine that
$\mathbf{D}=\mathbf{K}$, and

$\mathbf{\Gamma}=\left(\begin{array}[]{cc}\Gamma_{a}&\Gamma_{b\rightarrow
a}\\\ \Gamma_{a\rightarrow
b}&\Gamma_{b}\end{array}\right)=\left(\begin{array}[]{cc}\frac{|k_{a1}|^{2}+|k_{a2}|^{2}}{2}&k_{a2}k_{b2}^{*}e^{i\phi}\\\
k_{a1}^{*}k_{b1}e^{i\phi}&\frac{|k_{b1}|^{2}+|k_{b2}|^{2}}{2}\end{array}\right).$
(5)

Here, $\Gamma_{a}$, $\Gamma_{b}$ are the external damping rates into the
waveguide modes, while $\Gamma_{a\rightarrow b(b\rightarrow a)}$ is the
radiative interaction between the two resonators via waveguide.

##  III Transmission properties of the model

We now set out to show how to derive the transmission parameters for the
forward propagation ($\varepsilon_{1\rightarrow}$ drive) and the backward
propagation ($\varepsilon_{2\leftarrow}$ drive) from the basic equations in
Section II. In the long-time limit, the operators in the Hamiltonian [Eq. (1)]
reduce to their expectation values. For conciseness, we omit the notation
$\langle.\rangle$, and the terms $a$, $b$ and
$\varepsilon_{\ell\rightleftarrows}$ now denote the complex amplitudes of the
resonant and waveguide modes, respectively. When achieving the steady state
with $\dot{a}=0$, $\dot{b}=0$, Eq. (2) results in the nonlinear response of
the system to the counter-propagating drives,

$\displaystyle D(x)a$
$\displaystyle=\varepsilon_{1\rightarrow}[-k_{a1}(i\Delta_{b}+\gamma_{b}+\Gamma_{b})+k_{b1}e^{i\phi}(\Gamma_{b\rightarrow
a}+iJ)]$ (6)
$\displaystyle\\!\\!\\!\\!\\!\\!+\varepsilon_{2\leftarrow}[-k_{a2}e^{i\phi}(i\Delta_{b}+\gamma_{b}+\Gamma_{b})+k_{b2}(\Gamma_{b\rightarrow
a}+iJ)],$ $\displaystyle D(x)b$ (7)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\varepsilon_{1\rightarrow}[k_{a1}(\Gamma_{a\rightarrow
b}+iJ^{*})-k_{b1}e^{i\phi}(i\Delta_{a}+\gamma_{a}+\Gamma_{a}+2ix)]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\varepsilon_{2\leftarrow}[k_{a2}e^{i\phi}(\Gamma_{a\rightarrow
b}+iJ^{*})-k_{b2}(i\Delta_{a}+\gamma_{a}+\Gamma_{a}+2ix)],$

where

$\displaystyle D(x)$ $\displaystyle=(\Gamma_{a\rightarrow
b}+iJ^{*})(\Gamma_{b\rightarrow a}+iJ)$ (8)
$\displaystyle\\!\\!\\!\\!\\!\\!-(i\Delta_{a}+\gamma_{a}+\Gamma_{a}+2ix)(i\Delta_{b}+\gamma_{b}+\Gamma_{b}),$

and $x=U|a|^{2}$. The value of $x$ is obtained by solving the cubic equation
that results from taking the square of the norm of both sides of Eq. (6). This
may yield multiple solutions for $x$, which will be discussed in detail in
Section V.

The transmission parameters ($t_{\rightarrow}$ and $t_{\leftarrow}$) for both
forward and backward driving direction ($\varepsilon_{1\rightarrow}$ and
$\varepsilon_{2\leftarrow}$ drive, respectively) can be determined
analytically using the steady-state solutions Eq. (6), (7), and the input-
output relation Eq. (3). When we set the backward drive
$\varepsilon_{2\leftarrow}=0$, $x=x_{\rightarrow}$ is determined, the
transmission parameter for the forward (rightward) propagation from port $1$
is

$\displaystyle
t_{\rightarrow}=\frac{\varepsilon_{2\rightarrow}}{\varepsilon_{1\rightarrow}}$
$\displaystyle=\frac{e^{i\phi}}{D(x_{\rightarrow})}[iJ^{*}(\Gamma_{b\rightarrow
a}-\Gamma^{*}_{a\rightarrow b}+iJ)$ (9)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-(i\Delta_{a}+\gamma_{a}-\frac{|k_{a1}|^{2}-|k_{a2}|^{2}}{2}+2ix_{\rightarrow})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times(i\Delta_{b}+\gamma_{b}-\frac{|k_{b1}|^{2}-|k_{b2}|^{2}}{2})],$

and when we set the forward driving $\varepsilon_{1\rightarrow}=0$,
$x=x_{\leftarrow}$ is determined, the transmission parameter for the backward
(leftward) propagation from port $2$ is

$\displaystyle
t_{\leftarrow}=\frac{\varepsilon_{1\leftarrow}}{\varepsilon_{2\leftarrow}}$
$\displaystyle=\frac{e^{i\phi}}{D(x_{\leftarrow})}[iJ(\Gamma_{a\rightarrow
b}-\Gamma^{*}_{b\rightarrow a}+iJ^{*})$ (10)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-(i\Delta_{a}+\gamma_{a}+\frac{|k_{a1}|^{2}-|k_{a2}|^{2}}{2}+2ix_{\leftarrow})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times(i\Delta_{b}+\gamma_{b}+\frac{|k_{b1}|^{2}-|k_{b2}|^{2}}{2})].$

The parameter $x$ in the transmission parameter contains the effect of
nonlinearity.

When considering just one nonlinear resonator, $a$, with
$\phi=J=k_{b1}=k_{b2}=0$, the transmission parameters for both propagation
directions are reduced to

$t_{\rightarrow}=\frac{i(\Delta_{a}+2x_{\rightarrow})+\gamma_{a}-(|k_{a1}|^{2}-|k_{a2}|^{2})/2}{i(\Delta_{a}+2x_{\rightarrow})+\gamma_{a}+\Gamma_{a}},$
(11)
$t_{\leftarrow}=\frac{i(\Delta_{a}+2x_{\leftarrow})+\gamma_{a}+(|k_{a1}|^{2}-|k_{a2}|^{2})/2}{i(\Delta_{a}+2x_{\leftarrow})+\gamma_{a}+\Gamma_{a}}.$
(12)

The nonlinear response of the resonator $a$ to the input fields is

$a=\frac{k_{a1}\varepsilon_{1\rightarrow}+k_{a2}\varepsilon_{2\leftarrow}}{i(\Delta_{a}+2x)+\gamma_{a}+\Gamma_{a}}.$
(13)

When $|k_{a1}|=|k_{a2}|$, as required by time reversal symmetry [40, 41], the
system exhibits reciprocal transmission even if $U\neq 0$, as indicated in Eq.
(13) where $x$ will be the same for counter-propagating drives
$\varepsilon_{1\rightarrow}$ and $\varepsilon_{2\leftarrow}$. In other models
[35], a different direct scattering matrix $\mathbf{C}$ results in different
constraints when time reversal symmetry is imposed. This leads to
nonreciprocal behavior in systems with a single nonlinear resonator.

##  IV Is Nonreciprocity without Nonlinearity Possible?

We first investigate the system without any nonlinear effects, setting $U=0$,
thus $x=U|a|^{2}=0$. The presence of the second terms within parentheses in
Eq. (9) and Eq. (10) indicates that even in the absence of a direct coupling
$J$ between the resonators $a$ and $b$, nonreciprocity of the transmission can
manifest if $|k_{a1}|\neq|k_{a2}|$ or $|k_{b1}|\neq|k_{b2}|$. Conversely, If
$|k_{a1}|=|k_{a2}|$ and $|k_{b1}|=|k_{b2}|$, then the nonreciprocity condition
hinges on the condition that the coupling constant $J\neq 0$.

In the system with $|k_{a1}|=|k_{a2}|$ and $|k_{b1}|=|k_{b2}|$, from Eq. (9)
and Eq. (10), we can find that the condition for nonreciprocal transmission
($t_{\rightarrow}\neq t_{\leftarrow}$) is

$\displaystyle{\rm Re}[J(\Gamma_{a\rightarrow b}-\Gamma^{*}_{b\rightarrow
a})]\neq 0.$ (14)

For this system equipped with a real coupling constant $J$, the condition in
Eq. (14) is reduced to

${\rm Re}(\Gamma_{a\rightarrow b})\neq{\rm Re}(\Gamma_{b\rightarrow a}),$ (15)

which is determined by the phases of the coupling constants $k_{a\ell}$ and
$k_{b\ell}$, along with the phase shift $\phi$ accumulated between the two
resonators. When $k_{a\ell}$ and $k_{b\ell}$ are real, if $\cos\phi=0$, then
there is no nonreciprocity; if $\cos\phi\neq 0$, then the nonreciprocity
condition given by Eq. (15) simplifies to the requirement that
$k_{a1}k_{b1}\neq k_{a2}k_{b2}$. Since $|k_{a1}|=|k_{a2}|$ and
$|k_{b1}|=|k_{b2}|$, achieving nonreciprocity requires $k_{a1}=-k_{a2}$ and
$k_{b1}=k_{b2}$, or $k_{b1}=-k_{b2}$ and $k_{a1}=k_{a2}$ with
$\phi\neq(2N+1)\pi/2$ ($N\in\mathbb{Z}$). These are exactly the conditions
under which experiments in [6] are done.

In order to see how to achieve $k_{a1}=-k_{a2}$ or $k_{b1}=-k_{b2}$, let us
conside one of the resonators to be magnetic. For a magnetic resonator, the
coupling strength can flip its sign due to the reversal of the magnetic
field’s direction with counter-propagating directions. Most simply, a
monochromatic wave is governed by
$\mathbf{B}=\mathbf{k}\times\mathbf{E}/\omega$, where $\mathbf{k}$ denotes the
wave vector and $\omega$ denotes the wave frequency. The orientation of
$\mathbf{B}$ depends upon the direction of $\mathbf{k}$. Accordingly, the
potential energy of a magnetic dipole $\mathbf{m}$ in a magnetic field
$\mathbf{B}$, given by $E=-\mathbf{m}\cdot\mathbf{B}$, determines the sign of
the coupling strength between the magnetic resonator and the incoming waves.
Thus, when one of the resonators is magnetic, and the other is optical, i.e.,
$k_{a1}=-k_{a2}$ and $k_{b1}=k_{b2}$, or $k_{a1}=k_{a2}$ and $k_{b1}=-k_{b2}$,
with $\phi\neq(2N+1)\pi/2$, the nonreciprocity condition, as stated in Eq.
(15), is achieved [6].

In the system composed of two optical resonators where the coupling strengths
are equal for each resonator ($k_{a1}=k_{a2}$ and $k_{b1}=k_{b2}$), utilizing
the phase of the complex coupling strength $J$ can aid in achieving
nonreciprocal transmission. When $J$ is represented as $J=|J|e^{i\theta}$, the
nonreciprocal transmission condition can be achieved even when Eq. (15) is
violated, i.e., ${\rm Re}(\Gamma_{a\rightarrow b})={\rm
Re}(\Gamma_{b\rightarrow a})$. Specifically, setting $k_{a1}=k_{a2}=k_{a}$,
$k_{b1}=k_{b2}=k_{b}$, Eq. (9) and Eq. (10) can be reduced to

$t_{\rightarrow}=\frac{e^{i\phi}}{D(0)}[-2|J|k_{a}k_{b}e^{-i\theta}\sin\phi-|J|^{2}-(i\Delta_{a}+\gamma_{a})(i\Delta_{b}+\gamma_{b})],$
(16)
$t_{\leftarrow}=\frac{e^{i\phi}}{D(0)}[-2|J|k_{a}k_{b}e^{i\theta}\sin\phi-|J|^{2}-(i\Delta_{a}+\gamma_{a})(i\Delta_{b}+\gamma_{b})],$
(17)

which indicates that nonreciprocal transmission ($t_{\rightarrow}\neq
t_{\leftarrow}$) can occur when $\sin\theta\sin\phi\neq 0$, that is,
$\theta\neq N\pi$, $\phi\neq N\pi$. A recent paper discusses nonreciprocity
from complex $J$ within the context of magnetic films using the coherent
Dzyaloshinskii-Moriya (DM) interaction [25].

##  V Nonlinearity induced nonreciprocity if linearity does not allow
nonreciprocity

Figure 2: The $|\bar{\varepsilon}|^{2}$–$\tilde{x}$ curves, (a) for
$\varepsilon_{1\rightarrow}$ drive, (b) for $\varepsilon_{2\leftarrow}$ drive.
The turning points $\tilde{x}^{\pm}$ are as denoted in the figure. The black
lines are for the unstable states. $\Gamma/2\pi=10$ MHz,
$\tilde{\gamma}_{a}=\tilde{\gamma}_{b}=0.1$, $\tilde{\Delta}=-0.5$,
$\phi=16\pi/15$. Figure 3: Transmission spectrum as a function of the scaled
photon flux at the input $|\bar{\varepsilon}|^{2}$. The blue lines are for
$\varepsilon_{1\rightarrow}$ drive, and the yellow lines are for
$\varepsilon_{2\leftarrow}$ drive.
$\tilde{\gamma}_{a}=\tilde{\gamma}_{b}=0.1$. (a) $\tilde{\Delta}=0.5$,
$\phi=\pi/2$. (b) $\tilde{\Delta}=-0.5$, $\phi=16\pi/15$, consistent with Fig.
2. Sudden jumps in $|t|$ at $\tilde{x}^{-}$ are found as drives increase.
Figure 4: Nonreciprocity parameter
$I=(|t_{\rightarrow}|-|t_{\leftarrow}|)/(|t_{\rightarrow}|+|t_{\leftarrow}|)$
as a function of the phase shift $\phi$. The parameters are the same as Fig. 3
($\tilde{U}>0$). (a) $|\bar{\varepsilon}|^{2}=2$, $\tilde{\Delta}=0.5$. (b)
$|\bar{\varepsilon}|^{2}=0.9$, $\tilde{\Delta}=-0.5$. Figure 5: Nonreciprocity
parameter $I$ as a function of the phase shift $\phi$. Now we consider the
case when $\tilde{U}<0$. (a) $|\bar{\varepsilon}|^{2}=2$,
$\tilde{\Delta}=-0.5$. (b) $|\bar{\varepsilon}|^{2}=0.9$,
$\tilde{\Delta}=0.5$.

As discussed in Section IV, if the direct coupling $J$ is real and all
coupling constants are identical, then the linear medium fails to achieve
nonreciprocity. Here, we show how nonlinearity can produce significant
nonreciprocal behavior. We note that the nonlinearity induced nonreciprocity
has been studied before. Examples include intrinsic nonlinearities of two
level systems [10, 34]; nonlinearities of Brillouin media [8, 9]; nonlinear PT
symmetric media [44, 45, 46]; systems with a different scattering matrix
subjected to conditions imposed by time reversal symmetry [35, 36]. We now
consider a Kerr nonlinear system where there is no coherent coupling $J$
between the two resonators, i.e., $J=0$, and the coupling constants
$k_{a\ell}$, $k_{b\ell}$ are real and identical for counter-propagating
directions, i.e., $k_{a1}=k_{a2}$, $k_{b1}=k_{b2}$. However, the mode denoted
by $a$ is now characterized by third-order nonlinearity, i.e., $U\neq 0$. For
simplification, we also assume that $\Delta_{a}=\Delta_{b}=\Delta$,
$k_{a\ell}=k_{b\ell}=\sqrt{\Gamma}$, where $\Gamma$ denotes the total
radiative decay rate of an individual resonator into the waveguide port. At
the steady state with $\dot{a}=0$, $\dot{b}=0$, we have two equations for $a$
and $b$ followed from Eq. (2) and Eq. (3) ($a$, $b$ and
$\varepsilon_{\ell\rightleftarrows}$ denote the complex amplitudes as
discussed in Section III),

$\displaystyle
2i\tilde{U}|a|^{2}a+(i\tilde{\Delta}+\tilde{\gamma}_{a}+1)a+e^{i\phi}b=\tilde{\varepsilon}_{1\rightarrow}+e^{i\phi}\tilde{\varepsilon}_{2\leftarrow},$
(18)
$\displaystyle(i\tilde{\Delta}+\tilde{\gamma}_{b}+1)b+e^{i\phi}a=e^{i\phi}\tilde{\varepsilon}_{1\rightarrow}+\tilde{\varepsilon}_{2\leftarrow},$

along with two equations for the input-output relation,

$\displaystyle\tilde{\varepsilon}_{2\rightarrow}=e^{i\phi}\tilde{\varepsilon}_{1\rightarrow}-e^{i\phi}a-b,$
(19)
$\displaystyle\tilde{\varepsilon}_{1\leftarrow}=e^{i\phi}\tilde{\varepsilon}_{2\leftarrow}-a-e^{i\phi}b,$

all of which are normalized with the scaling factor $\Gamma$, leading to the
definitions: $\tilde{U}=U/\Gamma$, $\tilde{\Delta}=\Delta/\Gamma$,
$\tilde{\gamma}_{a}=\gamma_{a}/\Gamma$,
$\tilde{\gamma}_{b}=\gamma_{b}/\Gamma$,
$\tilde{\varepsilon}_{\ell\rightleftarrows}=\varepsilon_{\ell\rightleftarrows}/\sqrt{\Gamma}$.
From Eq. (18), the nonlinear response of resonator $a$ is obtained,

$\displaystyle 4\tilde{x}^{3}+4({\rm Re}\
\tilde{\delta})\tilde{x}^{2}+|\tilde{\delta}|^{2}\tilde{x}$ (20)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\pm\frac{|(i\tilde{\Delta}+\tilde{\gamma}_{b}+1-e^{2i\phi})\bar{\varepsilon}_{1\rightarrow}+(i\tilde{\Delta}+\tilde{\gamma}_{b})e^{i\phi}\bar{\varepsilon}_{2\leftarrow}|^{2}}{\tilde{\Delta}^{2}+(\tilde{\gamma}_{b}+1)^{2}},$

where we introduce $\tilde{x}=\tilde{U}|a|^{2}$,
$i\tilde{\delta}=i\tilde{\Delta}+\tilde{\gamma}_{a}+1-\frac{e^{2i\phi}}{i\tilde{\Delta}+\tilde{\gamma}_{b}+1}$,
$\bar{\varepsilon}_{1\rightarrow}=\sqrt{|\tilde{U}|}\tilde{\varepsilon}_{1\rightarrow}$,
$\bar{\varepsilon}_{2\leftarrow}=\sqrt{|\tilde{U}|}\tilde{\varepsilon}_{2\leftarrow}$,
with a plus sign in right hand side for $\tilde{U}>0$, and a minus sign for
$\tilde{U}<0$. In order to achieve nonreciprocity, in Eq. (20), the
coefficients before the input terms $\bar{\varepsilon}_{1\rightarrow}$ and
$\bar{\varepsilon}_{2\leftarrow}$ should differ, to ensure distinct nonlinear
responses to the counter-propagating drives. Nonreciprocity for the Kerr
nonlinear system then hinges on the condition $e^{2i\phi}\neq 1$, implying

$\phi\neq N\pi,\qquad(N\in\mathbb{Z}).$ (21)

Note that the cubic equation [Eq. (20)] can have three real roots under the
conditions

$({\rm Re}\ \tilde{\delta})^{2}>3({\rm Im}\
\tilde{\delta})^{2},\qquad\tilde{U}{\rm Re}\ \tilde{\delta}<0,$ (22)

leading to a bistability response [47]. Now we assmue $\tilde{U}>0$. We plot
$\tilde{x}$ against the scaled photon flux at the input
$|\bar{\varepsilon}|^{2}$ with experimentally feasible parameters which
satisfies Eq. (22), illustrated in Fig. 2. The
$|\bar{\varepsilon}|^{2}$–$\tilde{x}$ curve features two turning points
subject to $d(|\bar{\varepsilon}|^{2})/d\tilde{x}=0$. With specific region of
the amplitude of the input, there exist three real solutions for $\tilde{x}$,
with two representing stable states and one an unstable state. Fulfilling the
nonreciprocity criterion in Eq. (21) results in two distinct bistability
regions for counter-propagating drives, as illustrated in Fig. 2(a) and Fig.
2(b). As drives increase nearing the turning points $\tilde{x}^{-}$, sharp
jumps in $\tilde{x}$ from one stable state to another occur.

The transmission parameter can be obtained by substituting the expressions for
$a$ and $b$, derived from Eq. (18), into the input-output relation given by
Eq. (19). Once the value of $\tilde{x}$ is determined from Eq. (20) for
incoming drives ($\varepsilon_{1\rightarrow}$, $\varepsilon_{2\leftarrow}$)
from each direction, we can calculate out the value of $a$ from Eq. (18),

$a=\frac{(i\tilde{\Delta}+\tilde{\gamma}_{b}+1-e^{2i\phi})\tilde{\varepsilon}_{1\rightarrow}+(i\tilde{\Delta}+\tilde{\gamma}_{b})e^{i\phi}\tilde{\varepsilon}_{2\leftarrow}}{(i\tilde{\Delta}+\tilde{\gamma}_{a}+1+2i\tilde{x})(i\tilde{\Delta}+\tilde{\gamma}_{b}+1)-e^{2i\phi}},$
(23)

which exhibits distinct behaviours based on the propagation direction if Eq.
(21) is satisfied. When we set the backward driving
$\varepsilon_{2\leftarrow}=0$, the steady-state solution
$\tilde{x}=\tilde{x}_{\rightarrow}$ is determined from Eq. (20), the
transmission parameter for the forward (rightward) direction is

$t_{\rightarrow}=\frac{\varepsilon_{2\rightarrow}}{\varepsilon_{1\rightarrow}}=\frac{e^{i\phi}(i\tilde{\Delta}+\tilde{\gamma}_{a}+2i\tilde{x}_{\rightarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b})}{(i\tilde{\Delta}+\tilde{\gamma}_{a}+1+2i\tilde{x}_{\rightarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b}+1)-e^{2i\phi}}.$
(24)

Similarly, when we set the forward driving $\varepsilon_{1\rightarrow}=0$, the
steady-state solution $\tilde{x}=\tilde{x}_{\leftarrow}$ is determined from
Eq. (20), the transmission parameter for the backward (leftward) direction is

$t_{\leftarrow}=\frac{\varepsilon_{1\leftarrow}}{\varepsilon_{2\leftarrow}}=\frac{e^{i\phi}(i\tilde{\Delta}+\gamma_{a}+2i\tilde{x}_{\leftarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b})}{(i\tilde{\Delta}+\tilde{\gamma}_{a}+1+2i\tilde{x}_{\leftarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b}+1)-e^{2i\phi}}.$
(25)

Both transmission parameters share the same dependence on the parameter
$\tilde{x}$, which can be different between counter-propagating drives. When
the phase shift $\phi=N\pi$ (violating Eq. (21) and leading to
$\tilde{x}_{\rightarrow}=\tilde{x}_{\leftarrow}$), the system exhibits
reciprocal behavior, i.e., $t_{\rightarrow}=t_{\leftarrow}$. Nonetheless, with
$\phi\neq N\pi$, resulting in
$\tilde{x}_{\rightarrow}\neq\tilde{x}_{\leftarrow}$, it leads to
$t_{\rightarrow}\neq t_{\leftarrow}$, indicating nonreciprocal transmission.
The drive power $P$ is related to the coherent photon flux at the input,
through $P=\hbar\omega_{d}|\varepsilon|^{2}$, for the respective directions.
Eq. (20) shows that the value of $\tilde{x}$ depends on the scaled input,
i.e., $|\bar{\epsilon}|^{2}=UP/\hbar\omega_{d}\Gamma^{2}$. For a certain value
of $\tilde{x}$, the input power $P$ will be lower if $U$ becomes higher.

The nonreciprocal behavior of the system is illustrated in Fig. 3, where the
transmission amplitude $|t|$ is plotted against the scaled photon flux at the
input $|\bar{\epsilon}|^{2}$ for counter-propagating inputs. In Fig. 3(a), the
phase $\phi=\pi/2$ satisfies the nonreciprocity condition in Eq. (21), but
does not fulfill the bistability condition in Eq. (22). As a comparison, in
Fig. 3(b) with the same parameters as Fig. 2, the phase $\phi=16\pi/15$,
slightly detuned from $\pi$, satisfies both the nonreciprocity and bistability
conditions. As counter-propagating drives increases nearing $\tilde{x}^{-}$,
corresponding sudden jumps in the transmission amplitude $|t|$ within distinct
regions found in Fig. 3(b) due to bistability aids in achieving large
nonreciprocity. To measure the nonreciprocity, we define the nonreciprocity
parameter of the nonreciprocal system as
$I=(|t_{\rightarrow}|-|t_{\leftarrow}|)/(|t_{\rightarrow}|+|t_{\leftarrow}|)$.
This is graphically represented as a function of the phase shift $\phi$, with
a period of $\pi$, as depicted in Fig. 4. Additionally, we plot the
nonreciprocity parameter when $U<0$ as shown in Fig. 5.

In this section, both the resonators are considered optical. When the
resonator $a$ is switched to magnetic, i.e., $k_{a1}=-k_{a2}$ and
$k_{b1}=k_{b2}$, it results in merely a phase transformation,
$\phi\rightarrow\phi+\pi/2$, as discussed in Section VII.

##  VI Nonreciprocity in excitation of each resonator

Figure 6: The ratio of the excitation energy in nonlinear resonator $a$
($U>0$) relative to the input photon flux for counter-propagating drives as a
function of drive power. (a) $\tilde{\Delta}=0.5$, $\phi=\pi/2$. (b)
$\tilde{\Delta}=-0.5$, $\phi=16\pi/15$.

Driving the two resonators coherently from either the left-side or the right-
side give rise to the excitation energies in the resonators. For magnetic
resonators, the resulting excited magnetization can be electrically detected
through the spin currents via the inverse spin Hall effect (ISHE) [48]. For
the linear system ($U=0$), when driving from the left-side, i.e.,
$\varepsilon_{1\rightarrow}\neq 0$ and $\varepsilon_{2\leftarrow}=0$, the
energy ratio of the resonant mode $a$ relative to the incoming power is found
from Eq. (6) and Eq. (7),

$\displaystyle\left|\frac{a_{\rightarrow}}{\varepsilon_{1\rightarrow}}\right|^{2}$
(26)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1}{|D(0)|^{2}}(|k_{a1}|^{2}|i\Delta_{b}+\gamma_{b}+\Gamma_{b}|^{2}+|k_{b1}|^{2}|\Gamma_{b\rightarrow
a}+iJ|^{2})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\frac{1}{|D(0)|^{2}}[\Gamma_{a\rightarrow
b}(-i\Delta_{b}+\gamma_{b}+\Gamma_{b})(\Gamma_{b\rightarrow a}+iJ)+{\rm c.\
c.}].$

When driving from the right-side, i.e., $\varepsilon_{2\leftarrow}\neq 0$ and
$\varepsilon_{1\rightarrow}=0$, the energy ratio of the resonant mode $a$
relative to the incoming power is similarly found as

$\displaystyle\left|\frac{a_{\leftarrow}}{\varepsilon_{2\leftarrow}}\right|^{2}$
(27)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1}{|D(0)|^{2}}(|k_{a2}|^{2}|i\Delta_{b}+\gamma_{b}+\Gamma_{b}|^{2}+|k_{b2}|^{2}|\Gamma_{b\rightarrow
a}+iJ|^{2})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\frac{1}{|D(0)|^{2}}[\Gamma_{b\rightarrow
a}^{*}(-i\Delta_{b}+\gamma_{b}+\Gamma_{b})(\Gamma_{b\rightarrow a}+iJ)+{\rm
c.\ c.}].$

Nonreciprocal excitations can be achieved by ensuring asymmetric external
emission into counter-propagating waveguide modes, characterized by differing
coupling strengths, i.e., $|k_{a1}|\neq|k_{a2}|$ or $|k_{b1}|\neq|k_{b2}|$,
similar to nonreciprocal transmission as discussed in Section IV.

For a system where $|k_{a1}|=|k_{a2}|$ and $|k_{b1}|=|k_{b2}|$, the necessary
conditions to achieve nonreciprocal excitations
($\left|{a_{\rightarrow}}/{\varepsilon_{1\rightarrow}}\right|^{2}\neq\left|{a_{\leftarrow}}/{\varepsilon_{2\leftarrow}}\right|^{2}$)
are

$\Gamma_{a\rightarrow b}-\Gamma_{b\rightarrow a}^{*}\neq
0,\qquad\Gamma_{b\rightarrow a}+iJ\neq 0.$ (28)

When $k_{a\ell}$ and $k_{b\ell}$ are real, the first condition in Eq. (28) is
reduced to $\phi\neq N\pi$. This condition demonstrates that the reciprocity
of excitations can be broken through the manipulation of the phase shift
accumulated between two resonators.

For the Kerr nonlinear system with $\Delta_{a}=\Delta_{b}=\Delta$,
$k_{a\ell}=k_{b\ell}=\sqrt{\Gamma}$, and $J=0$, the necessary condition for
nonreciprocal exitations is $\phi\neq N\pi$. This condition, serving as the
basis for nonreciprocal transmission as discussed in Section V, can similarly
be extracted from Eq. (20). We plot the ratio
($|a/\tilde{\varepsilon}_{1\rightarrow}|^{2}$,
$|a/\tilde{\varepsilon}_{2\leftarrow}|^{2}$) for counter-propagating drives as
a function of the scaled photon flux as shown in Fig. 6.

Our Kerr nonlinear system can be configured with the resonator $a$ as either
optical type or magnetic type for experimental realization. For an optical
resonator, anharmonicity originates from nonlinear response of the electrical
polarization. Recent experimental advancements have showed cavities with
significant Kerr nonlinearity, measured by a Kerr coefficient of
$U/2\pi=-12.2\pm 0.1$ kHz/Photon [39]. For a magnetic resonator, anharmonicity
originates from the nonlinear magnetization. Recent research into
nonlinearities within ferrimagnetic spheres indicate that ferromagnetic
materials, such as Yttrium Iron Garnet (YIG), can exhibit robust coupling with
microwave fields against temperature. With the growing research interest in
YIG, it stands out as an active Kerr medium for exploring nonlinearity induced
nonreciprocity [49, 50].

##  VII Nonreciprocity Induced by Nonlinearity with a Magnetic Resonator

In this Section, We investigate the transmission property of the Kerr
nonlinear system, focusing on a case different from the one presented in
Section V. Here, the resonator $a$ is magnetic with third-order nonlinearity,
while the resonator $b$ is optical, characterized by different conditions
$k_{a1}=-k_{a2}$ and $k_{b1}=k_{b2}$. Similarly, there is no coherent coupling
$J$ between the two resonators ($J=0$), the coupling constants $k_{a\ell}$,
$k_{b\ell}$ are considered real. For convenience, we also assume
$\Delta_{a}=\Delta_{b}=\Delta$, $k_{a1}=-k_{a2}=\sqrt{\Gamma}$,
$k_{b\ell}=\sqrt{\Gamma}$. At the steady state with $\dot{a}=0$, $\dot{b}=0$,

$\displaystyle
2i\tilde{U}|a|^{2}a+(i\tilde{\Delta}+\tilde{\gamma}_{a}+1)a-e^{i\phi}b=\tilde{\varepsilon}_{1\rightarrow}-e^{i\phi}\tilde{\varepsilon}_{2\leftarrow},$
(29)
$\displaystyle(i\tilde{\Delta}+\tilde{\gamma}_{b}+1)b+e^{i\phi}a=e^{i\phi}\tilde{\varepsilon}_{1\rightarrow}+\tilde{\varepsilon}_{2\leftarrow},$

along with two equations for the input-output relation,

$\displaystyle\tilde{\varepsilon}_{2\rightarrow}=e^{i\phi}\tilde{\varepsilon}_{1\rightarrow}-e^{i\phi}a-b,$
(30)
$\displaystyle\tilde{\varepsilon}_{1\leftarrow}=e^{i\phi}\tilde{\varepsilon}_{2\leftarrow}+a-e^{i\phi}b,$

where $\tilde{U}$, $\tilde{\Delta}$, $\tilde{\gamma}_{a}$,
$\tilde{\gamma}_{b}$, $\tilde{\varepsilon}_{\ell\rightleftarrows}$ are as
defined in Section V. From Eq. (29), the nonlinear response of resonator $a$
is obtained,

$\displaystyle 4\tilde{x}^{3}+4({\rm Re}\
\tilde{\delta}^{\prime})\tilde{x}^{2}+|\tilde{\delta}^{\prime}|^{2}\tilde{x}$
(31)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{|(i\tilde{\Delta}+\tilde{\gamma}_{b}+1+e^{2i\phi})\bar{\varepsilon}_{1\rightarrow}-(i\tilde{\Delta}+\tilde{\gamma}_{b})e^{i\phi}\bar{\varepsilon}_{2\leftarrow}|^{2}}{\tilde{\Delta}^{2}+(\tilde{\gamma}_{b}+1)^{2}},$

where we define $\tilde{x}$, $\bar{\varepsilon}_{1\rightarrow}$,
$\bar{\varepsilon}_{2\leftarrow}$ as outlined in Section V, and introduce
$i\tilde{\delta}^{\prime}=i\tilde{\Delta}+\tilde{\gamma}_{a}+1+\frac{e^{2i\phi}}{i\tilde{\Delta}+\tilde{\gamma}_{b}+1}$,
with the assumption that $\tilde{U}>0$. When we set the backward driving
$\varepsilon_{2\leftarrow}=0$, the steady-state solution
$\tilde{x}=\tilde{x}_{\rightarrow}$ is determined from Eq. (31), which then
yields the transmission parameter for the forward (rightward) propagation,

$t_{\rightarrow}=\frac{\varepsilon_{2\rightarrow}}{\varepsilon_{1\rightarrow}}=\frac{e^{i\phi}(i\tilde{\Delta}+\tilde{\gamma}_{a}+2i\tilde{x}_{\rightarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b})}{(i\tilde{\Delta}+\tilde{\gamma}_{a}+1+2i\tilde{x}_{\rightarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b}+1)+e^{2i\phi}}.$
(32)

Similarly, when we set the forward driving $\varepsilon_{1\rightarrow}=0$, the
steady-state solution $\tilde{x}=\tilde{x}_{\leftarrow}$ is determined from
Eq. (31), which then yields the transmission parameter for the backward
(leftward) propagation,

$t_{\leftarrow}=\frac{\varepsilon_{1\leftarrow}}{\varepsilon_{2\leftarrow}}=\frac{e^{i\phi}(i\tilde{\Delta}+\gamma_{a}+2i\tilde{x}_{\leftarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b})}{(i\tilde{\Delta}+\tilde{\gamma}_{a}+1+2i\tilde{x}_{\leftarrow})(i\tilde{\Delta}+\tilde{\gamma}_{b}+1)+e^{2i\phi}}.$
(33)

Comparing the nonlinear response and the transmission parameters with those
discussed in Section V, the only crucial alteration is the transformation
$e^{2i\phi}\rightarrow-e^{2i\phi}$, equivalent to a phase shift,
$\phi\rightarrow\phi+\pi/2$. Consequently, in comparison to the case detailed
in Section V, switching the resonator $a$ from an optical to a magnetic type
results in a transformation in phase, thus in this case with no coherent
coupling ($J=0$), the system shows no nonreciprocity in the absence of
nonlinearity but would show significant nonreciprocity if the phase is not
equal to an odd multiple of $\pi/2$. With this transformation, the parameters
in Fig. 3(a) should be adjusted to $\phi=\pi$, and for Fig. 3(b), the
parameters should be adjusted to $\phi=47\pi/30$ to achieve corresponding
results.

For the linear system ($U=0$) equipped with a complex coupling constant
$J=|J|e^{i\theta}$, we consider the case when the coupling strengths are all
real, the nonreciprocal transmission ($t_{\rightarrow}\neq t_{\leftarrow}$)
can occur when $\cos\phi\cos\theta\neq 0$, that is, $\theta\neq(2N+1)\pi/2$,
$\phi\neq(2N+1)\pi/2$, which can be derived from Eq. (9) and Eq. (10).

##  VIII Conclusions

In our exploration of Kerr nonlinearity induced nonreciprocity in a system
with two waveguide-coupled resonators, we unveil the potential of Kerr
nonlinearity to manifest nonreciprocal behavior, particularly when linear
systems fail to exhibit nonreciprocal behavior. We derive the conditions
inducing nonreciprocity and study its dependency on various system parameters.
Our analysis reveals that nonreciprocal transmission is not only possible but
can be significantly enhanced through the strategic exploitation of
nonlinearity, thus in the case with no coherent coupling $J=0$, the system
shows no nonreciprocity in the absence of nonlinearity but can show
significant nonreciprocity with an appropriately chosen phase $\phi$.

Finally, it may be noted that we can discuss nonreciprocity in a full quantum
framework by converting the classical equation Eq. (2) into nonlinear quantum
Langevin equations for the mode operators. The very commonly used
linearization treatment of quantum fluctuations [51] gives quantum
fluctuations in terms of the values of the mean fields which are determined by
Eq. (2). Thus, the existence of nonreciprocal behavior of mean field equations
would imply nonreciprocity in quantum fluctuations as well. We hope to report
on these aspects in the future.

Acknowledgements

We thank the support of Air Force Office of Scientific Research (Award No
FA-9550-20-1-0366) and the Robert A Welch Foundation (A-1943-20210327). G. S.
A. thanks Prof. C. M. Hu and Y. Yang for the thesis of which enabled us to see
all the experiments on nonreciprocity in magnetic systems.

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