# Nonreciprocal Generation of Schrödinger Cat State Induced by Topology

Zi-Hao Li  School of Physics and Institute for Quantum Science and
Engineering, Huazhong University of Science and Technology, Wuhan 430074, P.
R. China  Li-Li Zheng  Key Laboratory of Optoelectronic Chemical Materials and
Devices of Ministry of Education, Jianghan University, Wuhan 430074, China
Ying Wu  School of Physics and Institute for Quantum Science and Engineering,
Huazhong University of Science and Technology, Wuhan 430074, P. R. China  Xin-
You Lü<EMAIL_ADDRESS>School of Physics and Institute for Quantum
Science and Engineering, Huazhong University of Science and Technology, Wuhan
430074, P. R. China

###### Abstract

The Schrödinger cat state produced differently in two directions is
anticipated to be a critical quantum resource in quantum information
technologies. By exploring the interplay between quantum nonreciprocity and
topology in a one-dimensional microcavity array, we obtain the Schrödinger cat
state (a pure quantum state) in a chosen direction at the edge cavity, whereas
a classical state in the other direction. This nonreciprocal generation of the
cat state originates from the topologically protected chirality-mode
excitation in the nontrivial phase, but in the trivial phase the nonreciprocal
generation of cat state vanishes. Thus, our proposal is switchable by tuning
the parameters so that a topological phase transition occurs. Moreover, the
obtained cat state has nonreciprocal high fidelity, nonclassicality, and
quantum coherence, which are sufficient to be used in various one-way quantum
technologies, e.g., invisible quantum sensing, noise-tolerant quantum
computing, and chiral quantum networks. Our work provides a general approach
to control quantum nonreciprocities with the topological effect, which
substantially broadens the fields of nonreciprocal photonics and topological
physics.

The Schrödinger cat state, as a macroscopic quantum superposition state, not
only plays a fundamental role in exploring the quantum-to-classical
transition, but also is crucial to quantum metrology Kira2011 ; Joo2011  and
fault-tolerant quantum computation Ralph2003 ; Mirrahimi2014 . The normal cat
states can be obtained by two approaches: (i) non-Gaussian measurement, which
typically using the technique called photon subtraction Nielsen2006 ; Sun2021
, and (ii) the nonlinear effect, e.g., single-photon Kerr effect Kirchmair2013
, optomechanical interaction Bose1997 , and so on Agarwal1997 ; Munoz2018 ;
Qin2021 . However, in previous schemes, these cat states are not prepared to
have difference in directions, thus cannot be used to apply one-way quantum
signal propagation, chiral quantum networks and invisible quantum sensing.
This suggests a strong demand of nonreciprocal generation of Schrödinger cat
states for the development of unidirectional quantum techniques Lodahl2017 .
The previous studies of optical nonreciprocities were primarily focus on the
nonreciprocal classical effects (e.g., the transmission of light), and various
methods have been proposed based on the spatial and temporal modulation of
material Yu2009 ; Sounas2017 , parity-time symmetry breaking Peng2014 ;
Chang2014 , etc.Hafezi2012 ; Shen2016 ; Maayani2018 ; Wang2019 ; Liang2020 ;
Tang2022 ; 2_Tang2022 . By contrast, research on quantum nonreciprocity is
still in the ascendant. Recently, by utilizing the existing techniques for
achieving classical nonreciprocity, purely quantum nonreciprocal effects such
as single-photon isolation Dong2021 , nonreciprocal photon, phonon or magnon
blockade Huang2018 ; Li2019 ; Yao2022 ; Wang2022 , and nonreciprocal
entanglement Jiao2020 ; Yang2020 ; Jiao2022 ; Liu2023  have been investigated.
Moreover, the nonreciprocal quantum correlations of photons can be achieved
even when no classical nonreciprocity exists Graf2022 ; Xiang2023 . This
suggests that the realization condition for quantum nonreciprocity might be
different from the condition for classical nonreciprocity. In particular, the
nonreciprocal generation of the Schrödinger cat state has not been proposed in
any system until now, since the previous proposals for controlling classical
nonreciprocities cannot be easily used to realize the nonreciprocal generation
of quantum states.

Figure 1: (Color online) (a) Scheme of nonreciprocal generation of the
Schrödinger cat states consists of a 1D microtoroid cavity array. Each cavity
supports two counter-circulating modes that are coupled with each other via a
scatterer, and the adjacent cavities are connected by the unidirectional
fibers. The cavity index number is denoted by $j$ below the cavity, $f_{j,\rm
R}$ and $\mathscr{F}_{\rm L}$ are the right propagating mode in the $j$th
purple fiber and the left propagating mode in the $j$th blue fiber,
respectively. The left-edge cavity in this array is a Kerr cavity with
nonlinearity $\chi$. The cavity array with zero nonlinearity $\chi=0$ can be
described by an SSH model shown at the right top. The dashed box is a unit
cell. $A$ and $B$ are two sublattices that assigned to the CW and CCW modes,
and $t_{1}$, $t_{2}$ refer to the on-site and off-site coupling strengths,
respectively. Given the same initial state $|\psi_{i}\rangle$ in the two modes
of the left-edge cavity, a high-fidelity Schrödinger cat state
$|\psi_{\circlearrowright}\rangle$ can be prepared in the CW mode, whereas a
classical state $|\psi_{\circlearrowleft}\rangle$ can be obtained in the
opposite direction. (b) Energy level of the edge cavity is shifted by the Kerr
nonlinearity. The blue and red solid arrow lines refer to the single-photon
and two-photon excitations, respectively. (c,d) The energy spectral of an open
array with 20 cavities and the corresponding winding number versus
$t_{1}/t_{2}$. The red lines in (c) mark the edge states.

The topological concept, originating from the study of solid-state electron
systems, has been extended to bosonic systems Haldane2008 ; Lu2014 ; Peano2015
; Huber2016 ; Tuo2019 ; Ozawa2019 ; Chen2019 ; Gao2020  and generates
substantial exotic quantum optics effects, including topologically protected
quantum coherence 2Wang2019 ; Nie2020 , topological lasing Harari2018 ;
Bandres2018 , and so on. Malzard2015 ; Bernardis2021 . In two-dimensional (2D)
topological systems, breaking time-reversal symmetry with a synthetic magnetic
field can result in the occurrence of topologically protected one-dimensional
(1D) chiral edge states propagating along multiple boundary sites Wang2008 ;
Goldman2012 . These chiral edge states are robust against disorder and immune
to backscattering, thus have been widely implemented in the optical
information processing Hafezi2011 ; Dlaska2017 , especially in the realization
of classical optical nonreciprocities, e.g., optical isolator and
unidirectional light propagation Mittal2014 ; Bahari2017 ; Solnyshkov2018 .
However, they have difficulty in forming the nonreciprocal generation of
single-site quantum states owing to their nonlocality. Moreover, the
nonreciprocal zero-dimensional (0D) topological edge state is still not
observed. In photonic systems such as microtoroid cavity arrays, the
bidirectional 0D edge naturally exists(the clockwise (CW) and counterclockwise
(CCW) traveling modes in one single cavity). Therefore, the topological
photonic systems provide a probability for finding the nonreciprocal 0D
topological edge state and then become a prospective platform for
nonreciprocal generation of the Schrödinger cat state.

Inspired by the above discussion, we present a scheme to realize the
topologically induced nonreciprocal generation of the Schrödinger cat state in
a 1D microtoroid cavity array. This 1D system is a photonic analog of the Su-
Schrieffer-Heeger (SSH) model, and there exists a topological phase transition
when moving the nanoparticles placed beside the cavities. Interestingly, the
nonreciprocal generation of the Schrödinger cat state is achieved in the edge
Kerr cavity when the 1D array is in the topological nontrivial phase.
Specifically, a high-fidelity Schrödinger cat state (quantum state) can be
prepared in the CW mode, but only a classical state can be prepared in the CCW
mode with the same initial condition. Physically, the topologically protected
chirality-mode excitation performed in two counter-propagating modes of the
edge cavity induces the appearance of the nonreciprocal Kerr effect, which
ultimately leads to the nonreciprocal generation of the Schrödinger cat state.
In particular, the nonreciprocal single-photon band structure in an analog SSH
coupled-cavity array has been proposed very recently Tang2022 . Different from
our topology-induced nonreciprocity, the nonreciprocal band structure in this
work is induced by the chiral interaction between the resonator array and
quantum emitters.

Moreover, we also show that the key properties of the obtained cat state,
including Wigner negativity, macroscopic quantum superposition, and fidelity
Sun2021 , also suggest strong nonreciprocity of cat state generation in the
topological nontrivial phase. This quantum nonreciprocity vanishes in the
trivial phase, which offers the important family of topological phase
transitions with a new type of application in controlling nonreciprocal
quantum resources. As far as we know, the nonreciprocal generation of the
Schrödinger cat state is proposed for the first time, which is not only
fundamentally interesting for quantum physics, but also has wide applications
ranging from one-way quantum information processing to constructing chiral
quantum networks. Our proposed topological-induced nonreciprocal mechanism in
our proposal is general, and can be extended to realize nonreciprocal
generation of other quantum states, e.g., Fock state and NOON states.

_System and nonreciprocal Kerr effect._ —We consider a 1D microtoroid cavity
array shown in Fig. 1(a), where every toroid cavity supports the CW and CCW
traveling modes with the same resonance frequency $\omega_{a}$. A nanoparticle
is placed in the mode volume of each cavity, and can induce a coupling between
the CW and CCW modes with strength $t_{1}$ by Rayleigh scattering. This
coupling can be controlled by manipulating the position of scatterer with a
nanopositioner Zhu2010 ; Peng2016 . The neighboring cavities are connected by
two unidirectional fibers Yu2009 , which enables a bidirectional coupling
between the CCW mode of the left cavity and the CW mode of the right cavity.
The edge cavity is a Kerr resonator, then the on-site Hamiltonian is

$\displaystyle H_{\rm c}=$
$\displaystyle\sum^{N}_{j=1}[\omega_{a}(a_{j,\circlearrowright}^{\dagger}a_{j,\circlearrowright}+a_{j,\circlearrowleft}^{\dagger}a_{j,\circlearrowleft})\\!+\\!(t_{1}a_{j,\circlearrowright}^{\dagger}a_{j,\circlearrowleft}+h.c.)]$
$\displaystyle+\chi(a_{1,\circlearrowright}^{\dagger}a_{1,\circlearrowright}^{\dagger}a_{1,\circlearrowright}a_{1,\circlearrowright}+a_{1,\circlearrowleft}^{\dagger}a_{1,\circlearrowleft}^{\dagger}a_{1,\circlearrowleft}a_{1,\circlearrowleft}),$
(1)

where $a_{j,\circlearrowright}(a_{j,\circlearrowleft})$ is the annihilation
operator of the CW (CCW) mode in the $j$th cavity. Here $\chi$ refers to the
Kerr nonlinear susceptibility, which can be introduced by various methods such
as Kerr material and electromagnetically induced transparency Schmidt1996 ;
Brasch2016 . In Fig. 1(b), we present the energy structure of a single Kerr
mode, with the Hamiltonian $H_{\rm s}=\omega_{a}a^{\dagger}a+\chi
a^{\dagger}a^{\dagger}aa$. Note that the first excited level is on-resonance,
whereas the second excited level is detuned by $2\chi$ owing to Kerr
nonlinearity. Corresponding to the two-photon excitation
$|0\rangle\rightarrow|2\rangle$, the frequency of the driving photon is
shifted by $\chi$.

The off-site link Hamiltonian can be written as

$\displaystyle\\!\\!H_{l}=$ $\displaystyle
i\sqrt{2\kappa_{f}}\sum^{N-1}_{j=1}[a_{j,\circlearrowleft}f^{\dagger}_{j,\rm
R}(t,z_{j})+a_{j,\circlearrowleft}\mathscr{F}^{\dagger}_{j,\rm L}(t,z_{j})$
$\displaystyle+a_{j+1,\circlearrowright}f^{\dagger}_{j,\rm
R}(t,z_{j+1})+a_{j+1,\circlearrowright}\mathscr{F}^{\dagger}_{j,\rm
L}(t,z_{j+1})\\!-\\!h.c.],$ (2)

where $\kappa_{f}$ is the loss induced by the resonator-fiber coupling,
$f_{j,\rm R}(t,z)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega
f_{j,\omega}e^{-i\omega(t-z/c)}$ is the right propagating fiber mode in the
$j$th purple fiber, and $\mathscr{F}_{\rm
L}(t,z)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega\mathscr{F}_{j,\omega}e^{-i\omega(t+z/c)}$
is the left propagating fiber mode in the $j$th blue fiber. Here, the $j$th
fiber is placed on the right of the $j$th cavity. Basically, the CCW mode of
the $j$th cavity can construct a two-way photon transition with the CW mode of
the ($j+1$)th cavity via the $j$th right and left propagating fibers. For the
CW mode of the $j$th cavity, it can construct a two-way photon transition with
the CCW mode of the ($j-1$)th cavity via the ($j-1$)th right and left
propagating fibers. As we will discuss later in this section, this is
important for building up a typical topological SSH structure in an arrayed
toroid cavity, where the CW and CCW modes become different sublattices of the
unit cell. We can obtain the quantum Langevin equations of every cavity modes,
and then eliminate the fiber modes to get the effective couplings between
cavities by using the Born-Markov approximation. The coupling coefficients are
chosen to be real by modifying the length of fibers, and the effective link
Hamiltonian thus becomes
$\tilde{H}_{l}=\sum^{N-1}_{j=1}(t_{2}a_{j,\circlearrowleft}^{\dagger}a_{j+1,\circlearrowright}+t_{2}a_{j+1,\circlearrowright}^{\dagger}a_{j,\circlearrowleft})$,
where $t_{2}$ is the coupling coefficient. In particular, the leftward hopping
($a_{j,\circlearrowleft}^{\dagger}a_{j+1,\circlearrowright}$) arises from the
blue fiber and the rightward hopping
($a_{j+1,\circlearrowright}^{\dagger}a_{j,\circlearrowleft}$) arises from the
purple fiber. As a result, the total Hamiltonian for the cavity array is
$H_{\rm tot}=H_{c}+\tilde{H}_{l}$.

Figure 2: (Color online) (a) Profiles of the edge state in the topological
nontrivial phase ($t_{1}/t_{2}=0.05$). The inset is the corresponding ground
state in the trivial phase ($t_{1}/t_{2}=2$). (b) Difference rate of the
chiral Kerr effect $R_{\rm k}$ versus $t_{1}/t_{2}$ when $\chi=4\pi\kappa$ and
$t_{2}=8\kappa$. (c,d) Excitation spectrum of the edge cavity with the present
of the Kerr nonlinearity $\chi=4\pi\kappa$ in the (c) topological phase
($t_{1}=0.8\kappa$, $t_{2}=16\kappa$) and (d) trivial phase ($t_{1}=8\kappa$,
$t_{2}=4\kappa$). The inset in (d) shows the edge cavity driven from opposite
directions. Other parameters are $N=5$, $\gamma=\kappa$, and
$\epsilon=8\kappa$.

For zero Kerr nonlinearity, i.e., $\chi=0$, the total Hamiltonian $H_{\rm
tot}$ describes an SSH model, where the CW and CCW modes serve as sublattices
A and B, respectively. According to the topological insulator theory
Asboth2016 , the system enters into the topological nontrivial phase when
$t_{1}<t_{2}$, which associates with the degeneration of energy spectra and
the occurrence of two edge states, as shown in Fig. 1(c). This process can
also be carried by the winding number. Typically, we make Fourier
transformation on cavity modes under the periodic boundary condition and
obtain
$a_{k,{\circlearrowright/\circlearrowleft}}=1/\sqrt{N}\sum_{j}a_{j,{\circlearrowright/\circlearrowleft}}e^{ikj}$.
Then we can write the Bloch Hamiltonian as $H_{\rm
tot}(\chi=0)=\sum_{k}V^{\dagger}H_{\rm tot}(k)V$, where
$V=(a_{k,{\circlearrowright}},a_{k,{\circlearrowleft}})^{T}$ and $H_{\rm
tot}(k)=\omega_{a}I+d_{x}\sigma_{x}+d_{y}\sigma_{y}$ with
$d_{x}=t_{1}+t_{2}\cos(k)$, $d_{y}=t_{2}\sin(k)$. Finally, the winding number
is defined as

$\displaystyle W=1/(2\pi i)\int_{-\pi}^{\pi}dk\frac{d\,{\rm ln}h(k)}{dk},$ (3)

where $h(k)=d_{x}-id_{y}$, and the results are plotted in Fig. 1(d). To
explore the property of edge states, we plot the profiles of one edge state in
the topological phase and the corresponding ground state in the trivial phase
in Fig. 2(a). Interestingly, in the topological phase, the CW mode of the
first cavity is occupied and the corresponding CCW mode is topologically
inhibited. In contrast, both the CW and CCW modes of the corresponding ground
state can be occupied when the system enters into the trivial phase by
adiabatically changing $t_{1}/t_{2}$. This leads to a topologically protected
chirality-mode-excitation in the single edge cavity when the microtoroid
cavity array is in the nontrivial phase. Based on the above discussion, we
emphasize that unidirectional parts such as one-way fibers are necessary for
the nonreciprocal generation of the Schrödinger cat state in this scheme. They
allow the CCW and CW modes in the $j$th cavity to couple with the CW mode of
the ($j+1$th) cavity and CCW mode of the ($j-1$th) cavity, respectively.
Consequently, the arrayed toroid cavity can be analogous to a typical
topological SSH model, and the symmetrical CW and CCW modes in one single
cavity can become different sublattices of the unit cell in the SSH structure.
This is crucial for topological-protected chiral-mode excitation on the CW and
CCW cavity modes in the edge cavity. Moreover, implementing such one-way
fibers is quite difficult in a periodic structure because it involves many
fibers. One should carefully adjust the direction and length of fibers as well
as guarantee that the fiber and cavity are correctly contacted in every unit
cell.

Figure 3: (Color online) Time evolution of the Wigner function of edge cavity
modes in (a) topological phase ($t_{1}=0.4\kappa$, $t_{2}=8\kappa$) and (b)
trivial phase ($t_{1}=6\kappa$, $t_{2}=4\kappa$). The upper row corresponds to
the CW mode and the lower row corresponds to the CCW mode. The inserted
$|\alpha|^{2}$ denotes the size of cat state. Other parameters are
$\alpha_{0}=2$, $\chi=2\pi\kappa$, and $N=5$.

Based on the above chiral mode excitation in the single edge cavity, the Kerr
nonlinear effect will exhibit great chirality in the nontrivial phase, even if
there is no difference between the Kerr strengths of the CW and CCW direction.
To show this chiral Kerr effect, we drive the edge cavity from the opposite
directions by a laser with power $P_{d}$ and frequency $\omega_{d}$, and
investigate the steady-state excitation spectrum of this cavity [see Fig.
2(d)]. The driven Hamiltonian is $H_{d}=i\sqrt{2\gamma}(\epsilon
e^{-i\omega_{d}t}a_{1,\circlearrowright/\circlearrowleft}^{\dagger}-h.c.)$,
where $\epsilon=\sqrt{2\pi P_{d}/\hbar\omega_{d}}$ and $\gamma$ is the loss
induced by the extra driving fiber. Based on the master equation,

$\displaystyle\dot{\rho}=$ $\displaystyle-i[H_{\rm
tot}+H_{d},\rho]+2\gamma\mathcal{L}(a_{1,\circlearrowright})\rho+2\gamma\mathcal{L}(a_{1,\circlearrowleft})\rho$
$\displaystyle+\sum_{j}(2\kappa\mathcal{L}(a_{j,\circlearrowright})\rho+2\kappa\mathcal{L}(a_{j,\circlearrowleft})\rho).$
(4)

we can obtain the steady-state density matrix $\rho_{s}$. Here, $\rho$ is the
density matrix of the total system, $\kappa$ is the total loss rate of the
cavity mode without external driving, and $\mathcal{L}(a)\rho=a\rho
a^{\dagger}-1/2\\{a^{\dagger}a,\rho\\}$. Then we can calculate the excitation
spectrum of the driven edge cavity as $\langle
n_{\circlearrowright/\circlearrowright}\rangle={\rm
Tr}[a^{\dagger}_{1,\circlearrowright/\circlearrowright}a_{1,\circlearrowright/\circlearrowright}\rho_{s}]$
with respect to different $\Delta$, as shown in Fig. 2(c). In the topological
phase, when the laser is driven from the CW direction, we can observed the
obvious peak at the Kerr sideband $\Delta=\omega_{a}-\omega_{d}=-\chi$
attributed to the two-photon resonance. This indicates the strong Kerr
nonlinearity of the CW mode. The peak at the Kerr sideband turns into two
separated peaks when the laser is driven from the CCW direction because the
driving from the opposite direction excites bulk states, which have
frequencies split away from the single-cavity resonant frequency. The
disappearance of the Kerr sideband peak indicates that the Kerr nonlinearity
of the CCW mode is strongly suppressed, then the chiral excitation at the Kerr
sideband $\Delta=-\chi$ can serve as an observable quantity for featuring the
chirality of the Kerr nonlinear effect. It is also shown in Fig. 2(d) that the
chiral Kerr effect between the CW and CCW modes almost vanishes in the trivial
phase. To view the detail variation of the chiral Kerr effect in the
topological and trivial phases, we define the corresponding difference rate as
$R_{\rm
k}=|n_{\circlearrowright}-n_{\circlearrowleft}|/(n_{\circlearrowright}+n_{\circlearrowleft})$,
where $n_{\circlearrowleft/\circlearrowright}$ is the CW/CCW mode excitation
number at $\Delta=-\chi$, and plot $R_{\rm k}$ as a function of $t_{1}/t_{2}$
in Fig. 2(b). The variation of these rates in the topological and trivial
phases further indicates that the topology can result in a great chiral Kerr
effect in the nontrivial phase.

_Nonreciprocal generation of the macroscopic quantum superposition state._
—The above nonreciprocal Kerr effect offers an opportunity for achieving the
nonreciprocal generation of the Schrödinger cat state. With the same initial
condition, we can prepared a high-fidelity Schrödinger cat state (quantum
state) in the CW mode but only a classical state in the CCW mode. To
investigate this nonreciprocal preparation, assume that the CW(CCW) mode on
the left-edge cavity is initially in a coherent state and the other cavity
field modes are initially in the vacuum states. To be specific, for the
preparation on the CW direction, the initial state is
$|\Psi_{\circlearrowright}(0)\rangle=|\alpha_{0}\rangle\otimes|0\rangle\otimes...\otimes|0\rangle$,
and for the preparation on the CCW direction, the initial state is
$|\Psi_{\circlearrowleft}(0)\rangle=|0\rangle\otimes|\alpha_{0}\rangle\otimes...\otimes|0\rangle$.
This is a natural assumption for coherent input light injected into the cavity
with a time scale shorter than $1/\kappa$. The time evolution of the system is
based on the $H_{\rm tot}$, which is described by the master equation

$\displaystyle\dot{\rho}=-i[H_{\rm
tot},\rho]+\sum_{j}(2\kappa\mathcal{L}(a_{j,\circlearrowright})\rho+2\kappa\mathcal{L}(a_{j,\circlearrowleft})\rho).$
(5)

In the topological nontrivial phase, the two states given by the time
evolution will exhibit significant differences between the opposite
directions. The numerical time evolutions of edge cavity modes in a 1D array
with 5 cavities are displayed in the left panel of Fig. 3. For the state
evolution of the CW mode, a three-component cat state with large size
$|\alpha|^{2}=2.9$ at time $\chi t=\pi/3$ and a two-component cat state with
large size $|\alpha|^{2}=2.5$ at time $\chi t=\pi/2$ are obtained.
Recognizable macroscopical peaks and their quantum interference can be
observed. These results can be explained by a single-mode time evolution with
the Kerr effect because the edge state in the topological phase is almost
completely localized on the CW mode of the edge cavity and the Kerr effect of
the CW mode is effectively excited. In this single-mode time evolution, the
initial state should be $|\alpha_{0}\rangle$, and the time evolution operator
is given by

$\displaystyle U(t)=exp(-i(\omega_{a}a^{\dagger}a+\chi
a^{\dagger}a^{\dagger}aa)t).$ (6)

In the interaction picture($\omega_{a}=0$), the $U(t)$ will give a state of
the form
$|\psi\rangle=e^{-it[(a^{\dagger}a)^{2}-a^{\dagger}a]\chi}|\alpha_{0}\rangle=e^{-|\alpha_{0}|^{2}/2}\sum_{n}{\alpha_{0}^{n}/\sqrt{n!}\;e^{-it(n^{2}-n)\chi}|n\rangle}$
at time $t$. When $\chi t=\pi/3$, the state will evolve to the three-component
cat state
$|\psi\rangle=c_{1}|\alpha_{0}e^{-i(2\pi/3)}\rangle+c_{2}|\alpha_{0}\rangle+c_{3}|\alpha_{0}e^{i(2\pi/3)}\rangle$,
with $c_{1}=(1-2e^{-i\pi/3})/3$, $c_{2}=c_{3}=(1+e^{-i\pi/3})/3$. At $\chi
t=\pi/2$, the state becomes
$|\psi\rangle=1/\sqrt{2}(e^{i\pi/4}|-i\alpha_{0}\rangle+e^{-i\pi/4}|i\alpha_{0}\rangle)$,
which is a two-component cat state. However, for the CCW case, the state
evolutions are apparently different, and only the states with low
nonclassicality are obtained at the same time. This can be explained by the
suppression of Kerr effect in the CCW mode owing to chiral-mode excitation.
Finally, the obtained state can be extracted through a quantum state
transferring protocol Xiang2017 . We note that if we inject the input light
and extract the final state by the same extra fiber coupled to the cavity,
then the input field in one direction will only result in the extracted state
in the opposite direction. And the same coherent input fields in the two
directions will result in different extracted states (a quantum state and a
classical state) after the same period of time. Thus our proposed model can
give the nonreciprocal generation of the cat state in the topological phase.
Furthermore, this quantum nonreciprocity reduces significantly in the trivial
phase. As shown in the right panel of Fig. 3, the generations of cat states
perform insignificant distinction for the CW and CCW modes at the same time.
For our chosen parameters, neither the CW nor CCW mode can evolve into a high-
quality cat state. This result can be understood as follows: the nonreciprocal
Kerr effect is substantially destructed when the microcavity array enters into
the trivial phase, and thus, the above quantum nonreciprocity vanishes.

In order to quantify the nonreciprocity of generation more clearly, we
calculate the figures of merit for the two-component cat states, and compare
their differences between the CW and CCW modes in Fig. 4. Here $\delta$ is the
Wigner negativity to evaluate the nonclassicality of the state. $I$ is used to
quantify the macroscopic quantum superposition, and a perfect cat state has
the maximum value $I=\langle a^{\dagger}a\rangle$. $F$ is the fidelity between
the actual state and the most similar cat state. To be specific, the most
similar cat state takes the form
$|\psi_{\sigma}(\eta,\beta,\gamma)\rangle=\mathcal{N}(|-\eta
e^{i\gamma}\rangle+e^{i\beta}|\eta e^{i\gamma}\rangle)$, where $\mathcal{N}$
is the normalization factor, $\eta$, $\beta$, and $\gamma$ are the fitting
coefficients. We can obtain $\eta$, $\beta$, and $\gamma$ from the
optimization calculation described by

$\displaystyle\mathop{\rm
min}_{\eta,\beta,\gamma}(\int[W_{|\psi_{\rho}\rangle}(\bm{\alpha})-W_{|\psi_{\sigma}(\eta,\beta,\gamma)\rangle}(\bm{\alpha})]^{2}d^{2}\bm{\alpha}),$
$\displaystyle
s.t.\quad\alpha\in[1.5,+\infty),\beta\in[-\pi,+\pi],\gamma\in[-\pi,+\pi],$ (7)

where $|\psi_{\rho}\rangle$ is the actual obtained state, and
$W_{|\psi\rangle}(\bm{\alpha})$ is the Wigner function of state
$|\psi\rangle$. Physically, the optimized state
$|\psi_{\sigma}(\eta,\beta,\gamma)\rangle$ has the most similar profile to the
actual state. The fidelity of the actually obtained state is measured with
respect to the most similar cat state instead of the theoretical state
$|\psi\rangle$ given by Eq. (6), because the dissipation in the realistic
environment will greatly modify the phase and size of the actually obtained
state. In some cases, the actually obtained state has little overlap with
$|\psi\rangle$, but still can be treated as the superposition of two
macroscopic coherent states. The Wigner negativity, macroscopic quantum
superposition, and fidelity are defined as

$\displaystyle\delta=\int(|W(\bm{\alpha})|-W(\bm{\alpha}))d^{2}\bm{\alpha},$
(8) $\displaystyle I=\frac{\pi}{2}\int
W(\bm{\alpha})(-\frac{\partial^{2}}{\partial\alpha\partial\alpha^{*}}-1)W(\bm{\alpha})d^{2}\bm{\alpha},$
(9) $\displaystyle F(\rho,\sigma)={\rm
Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\sim|\langle\psi_{\rho}|\psi_{\sigma}\rangle|,$
(10)

where $\rho$ and $\sigma$ are the density matrix of the obtained state
$|\psi_{\rho}\rangle$ and the most similar cat state $|\psi_{\sigma}\rangle$,
respectively. Firstly, it is shown from Fig. 4(a) that, in the topological
phase, the obtained two-component cat state for CW mode has high
nonclassicality ($\delta_{\circlearrowright}\approx 0.5$), fidelity
($F_{\circlearrowright}\to 1$), macroscopic quantum superpositions
($I_{\circlearrowright}\approx 2.6$), and large size ($|\alpha|^{2}\approx
2.5$), which are enough for practical applications in quantum technologies
Joo2011 ; Ralph2003 ; Mirrahimi2014 . Secondly, the above key properties of
the cat state also imply the strong nonreciprocity of the cat state generation
in the topological nontrivial phase. Comparing Fig. 4(a) and Fig. 4(c), one
can find that the nonclassicality is exhibited in the CW mode and vanishes in
the opposite direction. The Wigner negativities (dashed line) of the CW and
CCW modes are $\delta_{\circlearrowright}\approx 0.5$ and
$\delta_{\circlearrowleft}\approx 0.06$ at time $\chi t=\pi/2$, respectively.
The fidelities (solid line) of the CW and CCW modes are
$F_{\circlearrowright}\to 1$ and $F_{\circlearrowleft}\approx 0.73$ at time
$\chi t=\pi/2$, respectively, which indicates the great difference of the
quality of cat states obtained in the two directions. The different quantum
coherence signified by the macroscopic quantum superpositions $I$ is shown in
the inset with $I_{\circlearrowright}\approx 6I_{\circlearrowleft}$. A more
clear nonreciprocal rate can be defined based on fidelity as $R_{\rm
F}=|F_{\circlearrowright}-F_{\circlearrowleft}|$, and $R_{\rm F}=0.27$ is
found for the topological case. Lastly, it is shown from Fig. 4(b,d) that the
nonreciprocity implied by the above key properties of the cat state is
significantly reduced in the trivial phase. The Wigner negativity, macroscopic
quantum superpositions, and fidelity indicate that both the CW and CCW modes
cannot evolve into high-quality cat states in the trivial phase. Especially,
the fidelity for CW mode is $F_{\circlearrowright}\approx 0.7$ and for CCW
mode is $F_{\circlearrowleft}\approx 0.59$, the nonreciprocal rate is then
reduced to $R_{\rm F}=0.11$. This is attributed to the disappearance of chiral
mode excitation in the trivial phase. In the supplementary material, we show
that the nonreciprocal generation of a three-component cat state has similar
results.

Figure 4: (Color online) Time evolution of fidelity $F$, Wigner negativity
$\delta$, and macroscopic quantum superposition $I$ (inset) around $\chi
t=\pi/2$ in the topological phase (a,c), i.e., $t_{1}=0.4\kappa$,
$t_{2}=8\kappa$, and the trivial phase (b,d), i.e., $t_{1}=6\kappa$,
$t_{2}=4\kappa$. The inserted $|\alpha|^{2}$ in (a) indicates the size of
obtained cat state, and other parameters are the same as those in Fig. 3.

_Discussion and Conclusion._ —For experimental implementations, the resonator
array with an ultrahigh Q ($Q\sim 10^{8}$) microtoroid cavity has been
demonstrated in Refs. Armani2003 ; Ostby2009 . The required unidirectional
fibers that connect cavities are achievable through chiral interactions
Lodahl2017  or spatiotemporal modulation of material Yu2009 . Based on the
development of microcavity photonics, the strong Kerr nonlinearity for the
microresonator $\chi\approx 5$MHz has been achieved experimentally Palomo2017
; Zielinska2017 . Tunable backscattering interaction that can reach to
$t_{1}\approx 3$MHz was experimentally demonstrated by modifying the distance
between the scatterer and resonator Zhu2010 ; Peng2016 . Meanwhile, in the
$\rm Er^{3+}$-doped microcavity, a gain with rate $\mathcal{G}$ can be
introduced by optically pumping $\rm Er^{3+}$ ions with a pump laser, and thus
the effective loss rate $\kappa$ of the cavity mode becomes tunable Peng2014 ;
Chang2014 . In this case, the required coupling-loss relation $t_{1,2}/\kappa$
and nonlinearity-loss relation $\chi/\kappa$ can be achieved by employing the
proven $\rm Er^{3+}-$doping technologies in our proposal.

In summary, we have proposed an experimentally feasible scheme to realize the
nonreciprocal generation of the Schrödinger cat state, and revealed the
relation between this quantum nonreciprocity and topology in a 1D microtoroid
cavity array. This nonreciprocal generation of macroscopic quantum
superposition emerges in the edge cavity when the microcavity array enters
into the topological nontrivial phase. We have also calculated the key
properties of cat state, including the Wigner negativity, macroscopic quantum
superposition and fidelity, and all of them imply that the generations of cat
state exhibit unidirectional high qualities in the topological nontrivial
phase. This guarantees that the obtained nonreciprocal generation of the cat
state, as a pivotal quantum resource, can be used to implement one-way quantum
information processing with high efficiency. Our work opens up a door for
exploring a new type of nonreciprocal quantum resources, together with their
applications in quantum information science and photonics by utilizing the
topology of system.

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Supplemental Material for
“Nonreciprocal Generation of Schrödinger Cat State Induced by Topology”

Zi-Hao Li1, Li-Li Zheng2, Ying Wu1, Xin-You Lü1

1School of Physics and Institute for Quantum Science and Engineering, Huazhong
University of Science and Technology, Wuhan 430074, China
2Key Laboratory of Optoelectronic Chemical Materials and Devices of Ministry
of Education, Jianghan University, Wuhan 430074, China

This supplemental material contains six parts: I. A detailed derivation for
the effective link Hamiltonian $\tilde{H}_{l}$ used in the main text. II. An
additional analysis for nonreciprocal generation of three-component cat State.
III. Discussion of nonreciprocal performance for different number of cavities.
IV. An additional discussion for the influence of cross-Kerr nonlinearity. V.
Some theoretical analyses for the nonreciprocal transmission of optical field.
VI. Experimental feasibility of our proposal.

##  S1 Derivation of effective link Hamiltonian

In this section, we briefly describe the derivation of effective link
Hamiltonian $\tilde{H}_{l}$ by eliminating the fiber modes. We consider that
the $j$th cavity and the $(j+1)$th cavity are connected by two unidirectional
fibers, as shown in Figs. S1, and the link Hamiltonian reads

$\displaystyle\\!\\!H_{l}=$ $\displaystyle
i\sqrt{2\kappa_{f}}[a_{j,\circlearrowleft}f^{\dagger}_{j,\rm
R}(t,z_{j})+a_{j,\circlearrowleft}\mathscr{F}^{\dagger}_{j,\rm
L}(t,z_{j})+a_{j+1,\circlearrowright}f^{\dagger}_{j,\rm
R}(t,z_{j+1})+a_{j+1,\circlearrowright}\mathscr{F}^{\dagger}_{j,\rm
L}(t,z_{j+1})\\!-\\!h.c.],$ (S1)

where $\kappa_{f}$ is the loss induced by the resonator-fiber coupling, and
$z_{j}$ is the cavity position. Here, $f_{j,\rm R}(t,z)$ is the right
propagating fiber mode in $j$th purple fiber. $\mathscr{F}_{j,\rm L}(t,z)$ is
the left propagating fiber mode in $j$th blue fiber, where the $j$th fiber is
placed on the right of $j$th cavity. They are represented by a continuum of
left and right propagating bosonic mode, i.e.,
$f_{j,R}(t,z)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega
f_{j,\omega}e^{-i\omega(t-z/c)}$, $\mathscr{F}_{j,\rm
L}(t,z)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega\mathscr{F}_{j,\omega}e^{-i\omega(t+z/c)}$
and
$[f_{j,\omega},f^{\dagger}_{j,\omega^{\prime}}]=\delta(\omega-\omega^{\prime})$,
$[\mathscr{F}_{j,\omega},\mathscr{F}^{\dagger}_{j,\omega^{\prime}}]=\delta(\omega-\omega^{\prime})$.
In principle, it can be considered as two cascading systems Gardiner2004 ;
Bin2018 ; Cirac1997 ; Xiang2017 ; Stannigel2010 ; Stannigel2011 . We first
focus on the cascading system that describes the CCW mode in $j$th cavity and
CW mode in $(j+1)$th cavity connected by the right propagating fiber, and its
Hamiltonian is

$\displaystyle H_{\rm
R}=i\sqrt{2\kappa_{f}}(a_{j,\circlearrowleft}f^{\dagger}_{j,\rm
R}(t,z_{j})+a_{j+1,\circlearrowright}f^{\dagger}_{j,\rm R}(t,z_{j+1})-h.c.).$
(S2)

Figure S1: A 1D micro-toroid cavity array where each cavity supports two
counter-circulating modes that are coupled with each other via a scatterer.
The adjacent cavities are connected by the unidirectional fibers. The cavity
array can be described by a SSH model shown in the right top. The dashed box
is a unit cell that consists of the CW and CCW modes in one cavity. Here
$t_{1}$, $t_{2}$ are the on-site and off-site coupling strengths,
respectively. The nonreciprocal generation of Schrödinger cat states are
achieved in the left edge cavity with a Kerr nonlinearity $\chi$.

We can eliminate the fiber mode under the Born-Markov approximation, and
define the input and output field operators in the form $f^{\rm
in}_{j,R}(t,z_{j})=f_{R}(t,z_{j}+0^{-})$ and $f^{\rm
out}_{j,R}(t,z_{j})=f_{j,R}(t,z_{j}+0^{+})$, respectively. This gives the
dissipative dynamics in terms of quantum Langevin equation (QLE) of CCW mode
in $j$th and CW mode in ($j+1$)th cavity

$\displaystyle\dot{a}_{j,\circlearrowleft}$ $\displaystyle=$
$\displaystyle-\kappa_{f}a_{j,\circlearrowleft}-\sqrt{2\kappa_{f}}f_{j,\rm
R}^{\rm in}(t,z_{j}),$ $\displaystyle\dot{a}_{j+1,\circlearrowright}$
$\displaystyle=$
$\displaystyle-\kappa_{f}a_{j+1,\circlearrowright}-\sqrt{2\kappa_{f}}f_{j,\rm
R}^{\rm in}(t,z_{j+1}),$ (S3)

together with the input-output relation

$\displaystyle f_{j,\rm R}^{\rm out}(t,z_{j})$ $\displaystyle=$ $\displaystyle
f_{j,\rm R}^{\rm in}(t,z_{j})+\sqrt{2\kappa_{f}}a_{j,\circlearrowleft}(t),$
$\displaystyle f_{j,\rm R}^{\rm out}(t,z_{j+1})$ $\displaystyle=$
$\displaystyle f_{j,\rm R}^{\rm
in}(t,z_{j+1})+\sqrt{2\kappa_{f}}a_{j+1,\circlearrowright}(t).$ (S4)

In this cascading system, $f_{j,\rm R}^{\rm in}(t,z_{j})$ is a
$\delta$-correlated noise operator representing the white quantum noise. From
the unidirectional feature of the fiber, the output of $j$th CCW mode can
travel through fiber and become the input of the $(j+1)$th CW mode with a time
delay, which is described by the following form Stannigel2010 ; Stannigel2011

$\displaystyle f_{j,\rm R}^{\rm in}(t,z_{j+1})=f_{j,\rm
R}^{out}(t-\tau,z_{j})e^{i\omega\tau}.$ (S5)

Here, $\tau=(z_{j}-z_{j+1})/c$ is the propagation time between the neighboring
cavities. To obtain a simple form, we define the retard operators
$a_{j}(t)\rightarrow a_{j}(t-z_{j}/c)$. The term $e^{i\omega\tau}$ can be
settled to imaginary unit $i$ by tuning the positions of cavity along the
fiber to satisfy the relation $z_{j}-z_{j+1}=\pi c/(2\omega)$. As a result,
Eq. (S5) becomes

$\displaystyle f_{j,\rm R}^{\rm in}(t,z_{j+1})=if_{j,\rm R}^{\rm
out}(t,z_{j}).$ (S6)

For the cascading system that describes the CW mode in $j$th cavity and CCW
mode in $(j+1)$th cavity connected by the left propagating fiber, the relevant
Hamiltonian is

$\displaystyle H_{\rm
L}=i\sqrt{2\kappa_{f}}(a_{j,\circlearrowleft}\mathscr{F}^{\dagger}_{j,\rm
L}(t,z_{j})+a_{j+1,\circlearrowright}\mathscr{F}^{\dagger}_{j,\rm
L}(t,z_{j+1})-h.c.).$ (S7)

We can obtain the similar QLEs and input-output relations. We also consider
the input fields are quantum white noises written in the form

$\displaystyle f^{\rm in}_{j,R}(t,z_{j})=\mathscr{F}^{\rm
in}_{j,L}(t,z_{j+1})=dB(t),$ (S8)

with

$\displaystyle[dB(t)]^{2}=[dB^{\dagger}]^{2}=0,$ $\displaystyle
dB(t)dB(t)^{\dagger}=(N+1)dt,$ (S9) $\displaystyle[dB]^{\dagger}dB(t)=Ndt$

From this we can derive a master equation for density operator $\rho(t)$ of
the two cavity system by setting $\langle da(t)\rho\rangle\equiv\langle
ad\rho(t)\rangle$, and the master equation can be written in the form

$\displaystyle\dot{\rho}=$ $\displaystyle
2\kappa_{f}[\mathcal{L}(a_{j,\circlearrowright/\circlearrowleft})\rho+\mathcal{L}(a_{j+1,\circlearrowright/\circlearrowleft})\rho]+2\kappa_{f}\mathcal{L}_{\rm
noise}(a_{j,\circlearrowright/\circlearrowleft},a_{j+1,\circlearrowright/\circlearrowleft})\rho-2i\kappa_{f}([a_{j+1,\circlearrowright}^{\dagger},a_{j,\circlearrowleft}\rho]-[\rho
a_{j,\rm\circlearrowleft}^{\dagger},a_{j+1,\rm\circlearrowright}]).$ (S10)

Here, $\mathcal{L}(a)\rho=a\rho a^{\dagger}-1/2\\{a^{\dagger}a,\rho\\}$ is the
generalized Lindblad superoperator and $\mathcal{L}_{\rm
noise}(a,b)\rho=N/2([[a+b,\rho],a^{\dagger}+b^{\dagger}]+h.c.)$. The first
term of Eq. (S10) describes the decay of cavity introduced by cavity-fiber
coupling. In the case that $N=0$, which means the input fields are vacuum
fields, the second term can be safely eliminated. Then we can obtain a
stochastic Schrödinger equation for the cascaded systems from the master
equation. The evolution of stochastic wave function is written as

$\displaystyle\frac{d}{dt}|\Psi(t)\rangle=-iH_{l}^{\rm eff}|\Psi(t)\rangle,$
(S11)

where

$\displaystyle H_{l}^{\rm
eff}=(t_{2}a_{j,\rm\circlearrowleft}^{\dagger}a_{j+1,\rm\circlearrowright}+t_{2}a_{j+1,\rm\circlearrowright}^{\dagger}a_{j,\rm\circlearrowleft})$
(S12)

represents the effective cascading coupling between $j$th and $(j+1)$th cavity
and $t=2\kappa_{f}$ is the coupling coefficient. By counting all the cascading
couplings in the 1D cavity array, the effective link Hamiltonian of the total
system reads

$\displaystyle\tilde{H}_{l}=\sum^{N-1}_{j-1}(t_{2}a_{j,\rm\circlearrowleft}^{\dagger}a_{j+1,\rm\circlearrowright}+t_{2}a_{j+1,\rm\circlearrowright}^{\dagger}a_{j,\rm\circlearrowleft}).$
(S13)

##  S2 Additional analysis for the nonreciprocal generation of three-component
cat state

In the main text, it is shown that the nonreciprocal generation of three-
component Schrödinger cat states with large size can be achieved at $\chi
t=\pi/3$. Here we provide some detail analysis to quantify the characters
together with the nonreciprocity of the cat states generation by applying
three figures of merit, i.e., Wigner negativity $\delta$, macroscopic quantum
superpositions $I$ and fidelity $F$, defined as

$\displaystyle\delta=\int(|W(\bm{\alpha})|-W(\bm{\alpha}))d^{2}\bm{\alpha},$
(S14) $\displaystyle I=\frac{\pi}{2}\int
W(\bm{\alpha})(-\frac{\partial^{2}}{\partial\alpha\partial\alpha^{*}}-1)W(\bm{\alpha})d^{2}\bm{\alpha},$
(S15) $\displaystyle F(\rho,\sigma)={\rm
Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\sim|\langle\psi_{\rho}|\psi_{\sigma}\rangle|.$
(S16)

Figure S2: Time evolution of Fidelity $F$, Wigner negativity $\delta$ and
macroscopic quantum superposition $I$ (inset) around $\chi t=\pi/3$ in the
topological phase (a, c), i.e., $t_{1}=0.4\kappa$, $t_{2}=8\kappa$, and the
trivial phase (b, d), i.e., $t_{1}=6\kappa$, $t_{2}=4\kappa$. The inserted
$|\alpha|^{2}$ in (a) indicates the size of obtained cat state, and the other
parameters are $\alpha_{0}=2$, $\chi=2\pi\kappa$, $N=5$.

Here $W(\bm{\alpha})$ is the Wigner function of state, $\rho$ and $\sigma$ are
the density matrix of obtained state $|\psi_{\rho}\rangle$ and the most
similar cat state $|\psi_{\sigma}\rangle$, respectively. Fig S2(a) show that,
in topological nontrivial phase, the obtained three-component cat state with
size $|\alpha|^{2}=2.9$ in the CW mode at $\chi t=\pi/3$ approaches high
nonclassicality ($\delta_{\rm CW}\approx 0.9$), fidelity ($F_{\rm CW}\to 1$)
and macroscopic quantum superpositions ($I\approx 3$) that are enough for
practical applications. As predicted by the chirality mode occupation in the
topological phase, strong nonreciprocity appears when we compare two
counterpropagating modes, see Fig. S2(a) and S2(c). At time $\chi t=\pi/3$,
the difference of nonclassicality is revealed by the Wigner negativity (dashed
line) that reduces from $\delta_{\rm CW}\approx 0.9$ in CW mode to
$\delta_{\rm CCW}\approx 0.5$ in CCW mode. The quality of cat state in CW mode
is also obviously superior to that in CCW mode, as indicated by the Fidelity
(solid line) of CW $F_{\rm CW}\to 1$ and CCW modes $F_{\rm CCW}\approx 0.75$.
The macroscopic quantum superposition $I$ presented in the inset reaches
$I_{\rm CW}\approx 2I_{\rm CCW}$, showing the different quantum coherence of
counterpropagating modes. Compare to the two-component cat state, the
nonreciprocity of three-component cat state generation is slightly reduced
because of the short evolution time. In the trivial phase shown by Figs. S2(b)
and S2(d), the nonreciprocity almost disappears as displayed by the
insignificant difference of figures of merit between two counterpropagating
modes. This is due to the disappearance of chirality mode occupation.

##  S3 Comparation between different number of cavities

Figure S3: (Color online) (a) Energy spectrals of an open array with 6
cavities and 2 cavities (inset) versus $t_{1}/t_{2}$, The blue and yellow
background indicate the topological nontrivial and trivial region given by the
winding number of system. The red lines mark the edge modes. The black
triangles indicate that the edge modes at $t_{1}/t_{2}=0.4$ degenerate for 6
cavities but split for 2 cavities. (b) The profiles of edge states at
$t_{1}/t_{2}=0.4$ of an array with 6 cavities and 2 cavities (inset). (c)(d)
The Wigner function of the edge cavity states obtained at $\chi t=\pi/2$ after
the time evolution in an array with 2 cavities. (e)(f) The Wigner function
obtained in the same condition, but in an array with 6 cavities. (c)(e)
corresponds to the CW mode and the (d)(f) corresponds to the CCW mode. Other
parameters are $t_{1}=8\kappa$, $t_{2}=20\kappa$, $\alpha_{0}=2$,
$\chi=4\pi\kappa$.

In this section we compare the nonreciprocal preparation of two-component cat
state between different number of cavities. We first consider the cavity array
for zero Kerr nonlinearity. The energy spectras of an array with 6 cavities
and 2 cavities are plotted in Fig. S3(a). Here, the closed energy gap
corresponds to the appearance of edge mode. We can see that for both cases,
the energy gap close at $|t_{1}/t_{2}|<1$, and the closing point for 2
cavities is closer to zero point compare to the closing point for 6 cavities.
The closing point does not match the phase transition point given by the
winding number because the size of system is limited. The topology is a many-
body physical property of the system, thus in realistic system the topological
effect will diminish when the size of system is decreased. As we can see at
$t_{1}/t_{2}=0.4$, the closed energy gap for 6 cavities will open for 2
cavities. In Fig. S3(b), we plot the profiles of ground states at
$t_{1}/t_{2}=0.4$ in arrays with different sizes. In both arrays, we can see
that the CW mode dominate the left edge and the CCW mode dominate the right
edge. However, the chiral-mode-excitation at the edge cavity in an array with
6 cavities is much greater than that in an array with 2 cavities. This is a
clear signal that the chiral-mode-excitation in our model is topologically
protected, and the perfect chiral-mode-excitation is expected in a
sufficiently large array.

To compare the nonreciprocities of cat state generations at $t_{1}/t_{2}=0.4$
between arrays of different cavites, we simulate the numerical time evolutions
of the array with 2 cavities and 6 cavities following the master equation Eq.
(5) in the main text, and the Wigner functions of state obtained at time $\chi
t=\pi/2$ are displayed in Fig. S3(c)-(f). Here, the parameters are chosen as
$t_{1}=8\kappa$, $t_{2}=20\kappa$ and $\chi=4\pi\kappa$. Based on the
discussion in the main text, the system is expected to give the nonreciprocal
generation of cat state at this time. For arrays with different sizes, the
fidelies of states in CCW mode are $F=0.73$ for 2 cavities and $F=0.74$ for 6
cavities, which means the qualities of states are low and almost
indistinguishable for both cases. The difference can be found in CW mode,
where the fidelity of states for 6 cavities is $F=0.95$ and for 2 cavities the
fidelity is $F=0.83$. The quality for 6 cavities is obvioudly better than the
quality for 2 cavities. As a result, the nonreciprocity of cat state
generation for 6 cavities characterized by the nonreciprocal rate
$R_{F}(N=6)=0.12$ is higher than the nonreciprocity for 2 cavities
characterized by $R_{F}(N=2)=0.06$. The behavior of growing nonrecirpocity as
the increase of system size is clear evidence that this nonreciprocity is a
consequence of the topological effect.

##  S4 Influence of cross-kerr nonlinearity

Figure S4: (Color online) (a)(b) Wigner functions of the obtained edge cavity
states with cross-Kerr nonlinearity $\chi_{c}/\chi=0$. (c)(d) The Wigner
functions of the obtained edge cavity states with cross-Kerr nonlinearity
$\chi_{c}/\chi=2$. (a)(c) corresponds to the CW mode and the (b)(d)
corresponds to the CCW mode. (e) The nonreciprocal rates based on Fidelity of
obtained states $R_{\rm F}=|F_{\circlearrowright}-F_{\circlearrowleft}|$ with
different cross-Kerr nonlinearity. The inset show the Fidelities of obtained
state in CW and CCW modes. The lines represent the curve fitting of dots.
Other parameters are $t_{1}=0.4\kappa$, $t_{2}=8\kappa$, $\alpha_{0}=2$,
$\chi=2\pi\kappa$, $\chi t=\pi/2$.

Generally, the Kerr nonlinearity in the edge toroid cavity can be introduced
by various methods. However, some methods such as using optomechanical
coupling will cause the cross-Kerr nonlinearity between counter-propagating
modes Gong2009 ; Qian2021 . In this section, we discuss the influence of the
cross-Kerr nonlinearity on the nonreciprocal generation of quantum states. We
introduce the cross-Kerr Hamiltonian on the edge cavity of the form

$\displaystyle H_{\rm CK}=\chi_{\rm
c}a_{1,\circlearrowright}^{\dagger}a_{1,\circlearrowleft}^{\dagger}a_{1,\circlearrowright}a_{1,\circlearrowleft},$
(S17)

where $\chi_{\rm c}$ is the cross-Kerr nonlinear coefficient. In Fig. S4, we
numerically simulate the time evolutions following the master equation

$\displaystyle\dot{\rho}=-i[H_{\rm tot}+H_{\rm
CK},\rho]+\sum_{j}(2\kappa\mathcal{L}(a_{j,\circlearrowright})\rho+2\kappa\mathcal{L}(a_{j,\circlearrowleft})\rho),$
(S18)

and calculate the fidelities of obtained states for different cross-Kerr
nonlinear strength after the nonreciprocal generation processes. In Fig.
S4(a)(b), the Wigner functions of obtained states without cross-Kerr
nonlinearity are displayed. The clear difference of fidelities shows the
strong nonreciprocity of cat state generation in counter-propagating edge
cavity modes. When the cross-Kerr nonlinearity is introduced, the changed
Wigner functions of obtained states are displayed in Fig. S4(c)(d). Here, we
chose the cross-Kerr nonlinear strength to be twice as large as the self-Kerr
nonlinear strength. This is typically derived from the optomechanical system
(or cavity-atom system) where two optical modes are coupled to a common
mechanical oscillator (atom). The Wigner functions of obtained states show
that the cross-Kerr nonlinearity hardly change the fidelity in CW mode
($F_{\circlearrowright}=0.99$ for both $\chi_{c}/\chi=0$ and $\chi_{c}/\chi=2$
cases), but increase the fidelity in CCW mode ($F_{\circlearrowleft}=0.78$ for
$\chi_{c}/\chi=2$ and $F_{\circlearrowleft}=0.73$ for $\chi_{c}/\chi=0$). In
Fig. S4(e), we demonstrate the fidelities $F$ and nonreciprocal rate $R_{F}$
against the strength of cross-Kerr nonlinearity. The results also imply that
the cross-Kerr nonlinearity will reduce the nonreciprocity of cat state
generation mainly by enhancing the quality of obtained state in CCW mode. As
some self-Kerr nonlinearity generation methods accompany with cross-Kerr
nonlinearity may have potential superiority over other methods such as
feasibility in experiment, people can flexibly decide whether to adopt the
cross-Kerr nonlinearity according to their tolerance of nonreciprocal rate in
realistic situation. For example, if one can accept a nonreciprocal rate that
not less than 0.2, then they can adopt the cross-Kerr nonlinearity
$\chi_{c}/\chi=2$ caused by the optomechanical coupling or atom-resonator
coupling in our scheme. If they can’t accept this nonreciprocal rate, they can
use other approaches such as Kerr material and electromagnetically induced
transparency to generate self-Kerr nonlinearity without generating cross-Kerr
nonlinearity Schmidt1996 ; Brasch2016 .

##  S5 Additional discussion for nonreciprocal transmission

In the main text, we have shown the excitation spectrum at the Kerr sideband
to demonstrate the chiral Kerr effect. Actually, this chiral effect is related
to the nonreciprocal transmission rate of system. As a support for the main
text, in this section, we first analytically solve the transmission rate of
linear system, and give the numerical verification. Then we give the numerical
results of the transimission rate of the system in the presence of Kerr
nonlinearity. Through an extra fiber coupled to the left edge cavity [see Fig.
S5(a)], we drive the cavity from the opposite direction with power $P_{d}$ and
frequency $\omega_{d}$. For the linear system, the total Hamiltonian can be
written as $H_{\rm tot}=H_{c}+\tilde{H}_{l}+H_{d}$, where

$\displaystyle H_{\rm c}=$
$\displaystyle\sum^{N}_{j=1}[\omega_{a}(a_{j,\circlearrowright}^{\dagger}a_{j,\circlearrowright}+a_{j,\circlearrowleft}^{\dagger}a_{j,\circlearrowleft})\\!+\\!(t_{1}a_{j,\circlearrowright}^{\dagger}a_{j,\circlearrowleft}+h.c.)],$
(S19) $\displaystyle\tilde{H}_{l}=$
$\displaystyle\sum^{N-1}_{j}(t_{2}a_{j,\circlearrowleft}^{\dagger}a_{j+1,\circlearrowright}+t_{2}a_{j+1,\circlearrowright}^{\dagger}a_{j,\circlearrowleft}),$
(S20) $\displaystyle H_{d}=$ $\displaystyle i\sqrt{2\gamma}(\epsilon
e^{-i\omega_{d}t}a_{1,\circlearrowright/\circlearrowleft}^{\dagger}-h.c.).$
(S21)

Here, $\gamma$ is the loss induced by the extra driving fiber and
$\epsilon=\sqrt{2\pi P_{d}/\hbar\omega_{d}}$. The QLE of system with probe
field can be written in the matrix form
$\dot{a}=-Ma+u_{\circlearrowright/\circlearrowleft}$ where
$a=(a_{1,\circlearrowright},a_{1,\circlearrowleft},a_{2,\circlearrowright},a_{2,\circlearrowleft},\cdots)^{T}$,
$u_{\circlearrowright}=(\sqrt{2\gamma}\epsilon,...)^{T}$,
$u_{\circlearrowleft}=(0,\sqrt{2\gamma}\epsilon,...)^{T}$ and

$\displaystyle M$
$\displaystyle=\left(\begin{array}[]{ccccc}\gamma+\kappa+i\Delta_{p}&it_{1}&0&0&\cdots\\\
it_{1}&\gamma+\kappa+i\Delta_{p}&it_{2}&0&\cdots\\\
0&it_{2}&\kappa+i\Delta_{p}&it_{1}&\cdots\\\
0&0&it_{1}&\kappa+i\Delta_{p}&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right)$ (S22)

is a $N\times N$ matrix. Here, $\kappa$ is the loss induced by the coupling
between cavity and unidirectional fibers. The steady-state solution of
dynamical variables are given by
$\bar{a}_{\circlearrowright/\circlearrowleft}=M^{-1}u_{\circlearrowright/\circlearrowleft}$.

Figure S5: (a) Edge cavity is driven from the opposite direction through an
extra fiber coupled on the left side.The transmission rate $T$ of (b) linear
system and (c) system with the presence of Kerr nonlinearity $\chi=4\pi\kappa$
as a function of $t_{1}/t_{2}$. The solid lines denote the numerical results
and the crosses in (b) denote the analytical results of linear system. The
other parameters are $\gamma=\kappa$, $\epsilon=0.08\kappa$, $N=5$.

To obtain the inverse matrix of $M$, we start from the $(N-2)\times(N-2)$
matrix without the first two columns and rows, denoted as $D$. Writing $D$ in
block form Bernstein2005

$D=\left(\begin{array}[]{c|cccc}\kappa+i\Delta_{p}&it_{1}&0&0&\cdots\\\
\hline\cr it_{1}&\kappa+i\Delta_{p}&it_{2}&0&\cdots\\\
0&it_{2}&\kappa+i\Delta_{p}&it_{1}&\cdots\\\
0&0&it_{1}&\kappa+i\Delta_{p}&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right)=\left(\begin{array}[]{cc}E&F\\\ G&H\\\ \end{array}\right),$
(S23)

the inverse of matrix $D$ can be written as

$\displaystyle D^{-1}=\left(\begin{array}[]{cc}(E-FH^{-1}G)^{-1}&\cdots\\\
\vdots&\ddots\\\
\end{array}\right)=\left(\begin{array}[]{cc}(\kappa+i\Delta_{p}+t_{1}^{2}[H^{-1}]_{11})^{-1}&\cdots\\\
\vdots&\ddots\\\ \end{array}\right),$ (S24)

where $[H^{-1}]_{11}$ is the first row and first column in the inverse matrix
of $H$. The inverse of matrix $H$ can be obtained following the same step

$\displaystyle H^{-1}$
$\displaystyle=\left(\begin{array}[]{cc}(\kappa+i\Delta_{p}+t_{2}^{2}[R^{-1}]_{11})^{-1}&\cdots\\\
\vdots&\ddots\\\ \end{array}\right)$ (S25)

with

$R=\left(\begin{array}[]{ccccc}\kappa+i\Delta_{p}&it_{1}&0&0&\cdots\\\
it_{1}&\kappa+i\Delta_{p}&it_{2}&0&\cdots\\\
0&it_{2}&\kappa+i\Delta_{p}&it_{1}&\cdots\\\
0&0&it_{1}&\kappa+i\Delta_{p}&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right).$ (S26)

Here, $R$ is a $(N-4)\times(N-4)$ matrix. Comparing to the matrix $D$, $R$ has
the similar form, except that the first two columns and rows are missing.
Suppose that $N$ is large enough, we have $[D^{-1}]_{11}\approx[R^{-1}]_{11}$.
By combining Eq. (S24) and Eq. (S25), we get
$[D^{-1}]_{11}\approx[\kappa+i\Delta_{p}+t_{1}^{2}(\kappa+i\Delta_{p}+t_{2}^{2}[D^{-1}]_{11})^{-1}]^{-1}$
which yields

$\displaystyle[D^{-1}]_{11}=\frac{-t_{1}^{2}+t_{2}^{2}-(\kappa+i\Delta_{p})^{2}\pm\sqrt{4t_{2}^{2}(\kappa+i\Delta_{p})^{2}+(-t_{1}^{2}+t_{2}^{2}-(\kappa+i\Delta_{p})^{2})^{2}}}{2t_{2}^{2}(\kappa+i\Delta_{p})}.$
(S27)

Now we write the matrix $M$ in the form

$\displaystyle M$
$\displaystyle=\left(\begin{array}[]{cc|ccc}\gamma+\kappa+i\Delta_{p}&it_{1}&0&0&\cdots\\\
it_{1}&\gamma+\kappa+i\Delta_{p}&it_{2}&0&\cdots\\\ \hline\cr
0&it_{2}&\kappa+i\Delta_{p}&it_{1}&\cdots\\\
0&0&it_{1}&\kappa+i\Delta_{p}&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right)=\left(\begin{array}[]{cc}A&B\\\ C&D\\\ \end{array}\right).$
(S28)

The inverse of matrix $M$ can be written as

$\displaystyle M^{-1}=\left(\begin{array}[]{cc}(A-BD^{-1}C)^{-1}&\sim\\\
\sim&\sim\\\ \end{array}\right),$ (S29)

where

$\displaystyle A$
$\displaystyle=\left(\begin{array}[]{cc}\gamma+\kappa+i\Delta_{p}&it_{1}\\\
it_{1}&\gamma+\kappa+i\Delta_{p}\\\ \end{array}\right),$ (S30) $\displaystyle
BD^{-1}C$ $\displaystyle=\left(\begin{array}[]{cc}0&0\\\
0&-t_{2}^{2}[D^{-1}]_{11}\\\ \end{array}\right).$

Then we have

$\displaystyle\left[M^{-1}\right]_{11}=\frac{\gamma+\kappa+i\Delta_{p}+[D^{-1}]_{11}t_{2}^{2}}{t_{1}^{2}+(\gamma+\kappa+i\Delta_{p})(\gamma+\kappa+i\Delta_{p}+[D^{-1}]_{11}t_{2}^{2})},$
(S31)
$\displaystyle\left[M^{-1}\right]_{22}=\frac{\gamma+\kappa+i\Delta_{p}}{t_{1}^{2}+(\gamma+\kappa+i\Delta_{p})(\gamma+\kappa+i\Delta_{p}+[D^{-1}]_{11}t_{2}^{2})}.$

The steady-state value of $a_{1,\circlearrowright/\circlearrowleft}$ is

$\displaystyle\bar{a}_{1,\circlearrowright}$
$\displaystyle=\left[M^{-1}\right]_{11}\sqrt{2\gamma}\epsilon,$ (S32)
$\displaystyle\bar{a}_{1,\circlearrowleft}$
$\displaystyle=\left[M^{-1}\right]_{22}\sqrt{2\gamma}\epsilon.$ (S33)

For $\gamma=\kappa$, we derive the transmission rate of input field from
$T_{\circlearrowright/\circlearrowleft}=|1-\frac{\sqrt{2\gamma}}{\epsilon}\bar{a}_{1,\circlearrowright/\circlearrowleft}|^{2}=|t_{\circlearrowright/\circlearrowleft}|^{2}$
and obtain

$\displaystyle t_{\circlearrowright}$
$\displaystyle=\frac{(\Delta_{p}-2i\kappa)t_{1}^{2}+\Delta_{p}(t_{2}^{2}-(\Delta_{p}-i\kappa)(\Delta_{p}-3i\kappa)+A)}{\Delta_{p}t_{1}^{2}+(\Delta_{p}-2i\kappa)(t_{2}^{2}-(\Delta_{p}-i\kappa)(\Delta_{p}-3i\kappa)+A)},$
(S34) $\displaystyle t_{\circlearrowleft}$
$\displaystyle=\frac{\Delta_{p}t_{1}^{2}+(\Delta_{p}-2\kappa)(t_{2}^{2}-(\Delta_{p}-i\kappa)(\Delta_{p}+i\kappa)+A)}{\Delta_{p}t_{1}^{2}+(\Delta_{p}-2\kappa)(t_{2}^{2}-(\Delta_{p}-i\kappa)(\Delta_{p}-3i\kappa)+A)},$
(S35)

with

$\displaystyle
A=\sqrt{(t_{1}^{2}-(\Delta_{p}-t_{2}-i\kappa)^{2})(t_{1}^{2}-(\Delta_{p}+t_{2}-i\kappa)^{2})}.$
(S36)

In Fig. S5(b), we show the transmission rate of linear system for zero
detuning as a function of $t_{1}/t_{2}$. In topological phase ($t_{1}<t_{2}$),
the transmission rate of CW mode almost reach $0$ for a wide range, whereas
the transmission rate of CCW remain high, showing the obvious nonreciprocal
transmission rate of the edge cavity. This nonreciprocity reduce significantly
when the system enters into trivial phase ($t_{1}>t_{2}$), as the transmission
rate of CW mode increases drastically and becomes closer to the transmission
rate of CCW mode. The analytical results are roughly in agreement with
numerical results. The mismatch comes from: $(1)$ the finite array length in
numerical process and assumptive infinite array length in analytical process
and $(2)$ the missing of jump term in analytical QLE. In the presence of Kerr
nonlinearity, we plot the numerical results of transmission rate for zero
detuning in Fig. S5(c). Comparing to the chiral Kerr excitation showed in main
text, the change of nonreciprocal transmission in topological phase and
trivial phase is even more obvious. The nonreciprocal transmission of optical
field in topological phase also originates from the chirality mode occupations
in the edge cavity, it has wide applications in the engineering of optical
devices such as optical isolators, circulators and nonreciprocal single-photon
sources, as well as integrated optics and quantum network Caloz2018 ;
Lodahl2017 ; Jalas2013 .

##  S6 Discussion of experimental feasibility

As a support for the main text, here we discuss the experimental feasibility
of the chosen key system parameters in our proposal. In a microresonator, the
Kerr nonlinear parameter is $\chi=\hbar\omega_{a}^{2}cn_{2}/(n^{2}V_{\rm
eff})$, where $n_{2}(n)$ is the nonlinear(linear) refractive index, $c$ is the
speed of light in vacuum, $V_{\rm eff}$ is the nonlinear optical mode
Palomo2017 . According to the current accessible experimental conditions,
$V_{\rm eff}$ is typically $10–10^{3}\mu\rm m^{3}$ Vahala2003 ; Spillane2005 ,
$n_{2}\sim 10^{-14}$ can be realized in device constructed from potassium
titanyl phosphate crystal Zielinska2017 . With the resonator frequency
$\omega_{a}\sim 1$PHz and the quality factor $Q\sim 6\times 10^{8}$ Armani2003
; Ostby2009 , the dissipation rate $\kappa_{f}=\omega_{a}/Q\sim 1.6$MHz and
strong kerr nonlinearity for the microresonator $\chi\approx 3\kappa_{f}$ are
feasible to achieve. The backscattering coupling can be experimentally tuned
by changing the distance between the scatterer and resonator Zhu2010 ;
Peng2016  and reach $t_{1}\approx 1.5\kappa_{f}$. As derived in Sec. S1, the
off-site coupling is $t_{2}=2\kappa_{f}$. Moreover, we note the Kerr
nonlinearity can be enhanced by various approaches Lu2013 ; Bartkowiak2014 ;
Lu2015 ; Higo2018 . Meanwhile, the coupling-loss relation and nonlinearity-
loss relation can be achieved by changing the total loss rate $\kappa$. For
example, in the resonator made from $\rm Er^{3+}$-doped silica, a gain rate
$\mathcal{G}$ can be introduced by the optically pumping $\rm Er^{3+}$ ions
with a pump laser Peng2014 . For gain rate chosing around
$\mathcal{G}/\kappa_{f}\in[0,0.9]$, the total loss rate can be reduced to
$\kappa/\kappa_{f}=(\kappa_{f}-\mathcal{G})/\kappa_{f}\in[0.1,1]$, thus the
coupling-loss relations $t_{1}/\kappa\in[0,15]$, $t_{2}/\kappa\in[2,20]$ and
nonlinearity-loss relation $\chi/\kappa\in[3,30]$ in our proposal could be
achieved with feasible experimental parameters.

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