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# Quantum Squeezing Induced Optical Nonreciprocity

Lei Tang  College of Engineering and Applied Sciences, National Laboratory of
Solid State Microstructures, and Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210093, China  Jiangshan Tang
College of Engineering and Applied Sciences, National Laboratory of Solid
State Microstructures, and Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210093, China  Mingyuan Chen
College of Engineering and Applied Sciences, National Laboratory of Solid
State Microstructures, and Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210093, China  Franco Nori  RIKEN
Quantum Computing Center, RIKEN Cluster for Pioneering Research, Wako-shi,
Saitama 351-0198, Japan  Physics Department, The University of Michigan, Ann
Arbor, Michigan 48109-1040, USA  Min Xiao  College of Engineering and Applied
Sciences, National Laboratory of Solid State Microstructures, and
Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China  Department of Physics, University of
Arkansas, Fayetteville, Arkansas 72701, USA  Keyu Xia<EMAIL_ADDRESS>College of Engineering and Applied Sciences, National Laboratory of Solid
State Microstructures, and Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210093, China  Jiangsu Key
Laboratory of Artificial Functional Materials, Nanjing University, Nanjing
210023, China

###### Abstract

We propose an all-optical approach to achieve optical nonreciprocity on a chip
by quantum squeezing one of two coupled resonator modes. By parametric pumping
a nonlinear resonator unidirectionally with _a classical coherent field_ , we
squeeze the resonator mode in a selective direction due to the phase-matching
condition, and induce a chiral photon interaction between two resonators.
Based on this chiral interresonator coupling, we achieve an all-optical diode
and a three-port quasi-circulator. By applying _a second squeezed-vacuum
field_ to the squeezed resonator mode, our nonreciprocal device also works for
single-photon pulses. We obtain an isolation ratio of
$>40~{}\mathrm{d}\mathrm{B}$ for the diode and fidelity of $>98\%$ for the
quasi-circulator, and insertion loss of $<1~{}\mathrm{d}\mathrm{B}$ for both.
We also show that nonreciprocal transmission of strong light can be switched
on and off by a relative weak pump light. This achievement implies a
nonreciprocal optical transistor. Our protocol opens up a new route to achieve
integrable all-optical nonreciprocal devices permitting chip-compatible
optical isolation and nonreciporcal quantum information processing.

Optical nonreciprocal devices, such as optical diodes and circulators, can
separate backscattering signals from a light source. The conventional magneto-
optical approach to achieve optical nonreciprocity (ONR) is difficult to
integrate on a chip because it requires strong magnetic fields and bulky
materials. Developing a new mechanism for magnetic-free ONR is of interest in
fundamental physics and promises important applications for on-chip light
manipulation. Various magnetic-free optical nonreciprocal devices have been
theoretically proposed and experimentally demonstrated by exploiting optical
nonlinearities [1, 2, 3, 4, 5, 6, 7, 8, 9], spatio-temporal modulation of the
medium [10, 11, 12], spin-momentum locking in chiral quantum optical systems
[13, 14, 15, 16, 17, 18, 19, 20, 21], directional optomechanical coupling [22,
23, 24], moving atomic lattices [25, 26, 27], atomic reservoir engineering
[28], the Sagnac effect in spinning resonators [29, 30, 31], and
susceptiblity-momentum locking in atomic gases [32, 32, 33, 34, 35, 36, 37,
38]. Optical nonreciprocal devices based on the Kerr nonlinearity of materials
are compatible with a chip, but subject to dynamic reciprocity [4]. Despite
many efforts, it is challenging to realize a chip-compatible all-optical
nonreciprocal device without moving parts or spatio-temporal modulation. By
coherently driving a $\chi^{(2)}$ microring resonator, one can directionally
amplify the single-photon interaction in the resonator via the mode mean
field. This mechanism has been exploited to induce chiral normal mode
splitting (NMS) and construct on-chip optical isolation [39]. Nevertheless,
the demanded three-mode phase matching in one resonator is a significant
challenge, as pointed out in Ref. [40].

Resonator mode squeezing via parametric driving of a $\chi^{(2)}$ resonator
has been explored to exponentially amplify the interaction between quantum
objects [41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. However, it has not been
used to achieve ONR. Here, we show that high-performance ONR can be achieved
by directionally squeezing the resonator mode with a _coherent_ laser field.
With this chiral quantum squeezing, we achieve an optical diode, a quasi-
circulator and, for the first time, a nonreciprocal optical transistor. Note
that our method is based on directional quantum squeezing and thus
conceptually differs from Ref. [39]. Our method only needs two-mode matching
in one resonator and thus greatly simplifies its experimental implementation.

Figure 1: Schematic of an all-optical nonreciprocal system consisting of two
microring resonators and two nearby optical waveguides. To achieve _classical
light isolation_ , a coherent pump field is applied to generate a CCW
squeezing mode $b_{s_{\circlearrowleft}}$ in $\text{R}_{\text{B}}$. To achieve
_single-photon isolation_ , a broadband squeezed-vacuum field is used to drive
$\text{R}_{\text{B}}$. (a) A forward-input signal field excites a CW mode
$a_{{}_{\circlearrowright}}$ in $\text{R}_{\text{A}}$, which interacts with
the squeezed mode $b_{s_{\circlearrowleft}}$ with a coupling rate $J_{s}$. (b)
A backward-input signal field stimulates a CCW mode
$a_{{}_{\circlearrowleft}}$ in $\text{R}_{\text{A}}$. It couples to a CW bare
mode $b_{{}_{\circlearrowright}}$ in $\text{R}_{\text{B}}$ with a rate $J$.

The schematic of our proposed system is depicted in Fig. 1. It consists of two
coupled Whispering-Gallery mode microring resonators and two nearby optical
waveguides. The resonators can be made of high-quality thin-film with
$\chi^{(2)}$ nonlinearity, e.g., lithium niobate or aluminum nitride. Thus,
the resonators supports the parametric nonlinear optical process. The
resonator B ($\text{R}_{\text{B}}$) is pumped from port $3$ by a continuous
wave coherent laser field with frequency $\omega_{p}$, amplitude $\alpha_{p}$,
and phase $\theta_{p}$. This _classical_ pump generates a squeezing
interaction with strength $\Omega_{p}$ for the counterclockwise (CCW) mode
$b_{{}_{\circlearrowleft}}$. Because of the directional phase-matching
condition in the parametric nonlinear process, the mode
$b_{{}_{\circlearrowleft}}$ is squeezed to a mode $b_{s_{\circlearrowleft}}$,
but the clockwise (CW) mode $b_{{}_{\circlearrowright}}$ is unsqueezed. The
resonator A ($\text{R}_{\text{A}}$) slightly differs in size from
$\text{R}_{\text{B}}$, such that the pump field cannot drive the parametric
nonlinear process in the former. In this arrangement, we only need to consider
mode squeezing in $\text{R}_{\text{B}}$. When a signal inputs to port 2, it
“sees” a normal two-resonator system.

Now we discuss how the pump laser modulates the inter-resonator interaction
and causes the ONR. In the forward-input case, a signal field input to port
$1$ excites the CW mode $a_{{}_{\circlearrowright}}$ in $\text{R}_{\text{A}}$.
Because of the pump field, the mode $a_{{}_{\circlearrowright}}$ couples to
the squeezed mode $b_{s_{\circlearrowleft}}$ with a rate $J_{s}$. The
parametric nonlinear process related to $\Omega_{p}$ also causes a frequency
shift to $b_{s_{\circlearrowleft}}$ with respect to the bare mode
$b_{{}_{\circlearrowleft}}$. In comparison, in the backward-input case, a
signal field from port $2$ excites the CCW mode $a_{{}_{\circlearrowleft}}$ in
$\text{R}_{\text{A}}$. In this case, the pump field has no action on the CW
mode $b_{{}_{\circlearrowright}}$. This case is equivalent to the unpumped
system only consisting of two coupled resonators. Thus, the mode
$a_{{}_{\circlearrowleft}}$ interacts with the bare mode
$b_{{}_{\circlearrowright}}$ with a unmodulated coupling rate $J$, rather than
coupling to the squeezed mode. Because the opposite input signal “sees”
different system configurations, we can achieve ONR.

In the frame rotating at frequency $\omega_{p}/2$, the Hamiltonian for the
forward-input case reads

$\displaystyle\mathcal{H}_{\text{fw}}=\mathcal{H}_{\text{A}}+\mathcal{H}_{\text{B}}+\mathcal{H}_{J}\;,$
(1)
$\displaystyle\mathcal{H}_{\text{A}}/\hbar=\Delta_{p}^{a}a_{{}_{\circlearrowright}}^{\dagger}a_{{}_{\circlearrowright}}+i\sqrt{2\kappa_{\text{ex1}}}(a_{\text{in}}a_{{}_{\circlearrowright}}^{\dagger}e^{-i\Delta_{\text{in}}t}-a_{\text{in}}^{\dagger}a_{{}_{\circlearrowright}}e^{i\Delta_{\text{in}}t})\;,$
$\displaystyle\mathcal{H}_{\text{B}}/\hbar=\Delta_{p}^{b}b_{{}_{\circlearrowleft}}^{\dagger}b_{{}_{\circlearrowleft}}+\frac{\Omega_{p}}{2}(e^{-i\theta_{p}}b_{{}_{\circlearrowleft}}^{\dagger
2}+e^{i\theta_{p}}b_{{}_{\circlearrowleft}}^{2})\;,$
$\displaystyle\mathcal{H}_{J}/\hbar=J(a_{{}_{\circlearrowright}}^{\dagger}b_{{}_{\circlearrowleft}}+a_{{}_{\circlearrowright}}b_{{}_{\circlearrowleft}}^{\dagger})\;,$

where the detunings are $\Delta_{p}^{a/b}=\omega_{a/b}-\omega_{p}/2$ and
$\Delta_{\text{in}}=\omega_{\text{in}}-\omega_{p}/2$, $\omega_{a/b}$ is the
resonance frequency of $\text{R}_{\text{A/B}}$, $\omega_{\text{in}}$ is the
frequency of the signal mode $a_{\text{in}}$, and $\kappa_{\text{ex1}}$ is the
external decay rate of $\text{R}_{\text{A}}$. Experimentally, the pump field
$\Omega_{p}$ is created by driving a mode $c_{{}_{\circlearrowleft}}$ of
$\text{R}_{\text{B}}$ with an external field $\alpha_{p}$. To a good
approximation, the dynamics of mode $c_{{}_{\circlearrowleft}}$ with resonance
frequency $\omega_{c}$ can be modeled as a cavity driven by a laser with power
$P_{p}$ and frequency $\omega_{p}\approx 2\omega_{b}$ via the Hamiltonian
$H_{\text{p}}=\Delta_{p}^{c}c_{{}_{\circlearrowleft}}^{\dagger}c_{{}_{\circlearrowleft}}+i\sqrt{2\kappa_{\text{ex2}}^{p}}\alpha_{p}(c_{{}_{\circlearrowleft}}^{\dagger}-c_{{}_{\circlearrowleft}})$,
where $\alpha_{p}=\sqrt{2\pi P_{p}/\hbar\omega_{p}}$,
$\Delta_{p}^{c}=\omega_{c}-\omega_{p}$ and $\kappa_{\text{ex2}}^{p}$ is the
external decay rate of the mode $c_{{}_{\circlearrowleft}}$. In steady state,
we obtain $\langle
c_{{}_{\circlearrowleft}}\rangle_{\text{ss}}=\sqrt{2\kappa_{\text{ex2}}^{p}}\alpha_{p}\big{/}(i\Delta_{p}^{c}+\kappa_{p})$,
where $\kappa_{p}$ is the total decay rate of the mode
$c_{{}_{\circlearrowleft}}$. In the mean-field approximation, we have
$\Omega_{p}e^{-i\theta_{p}}=2g\langle
c_{{}_{\circlearrowleft}}\rangle_{\text{ss}}$, where $g$ is the single-photon
nonlinear coupling strength of the parametric nonlinear process (see
Supplemental Material [51]).

Applying the Bogoliubov squeezing transformation [52, 53, 41, 42]
$b_{s}=\cosh(r_{p})b+e^{-i\theta_{p}}\sinh(r_{p})b^{\dagger}$ with the
squeezing parameter $r_{p}=\frac{1}{4}\ln(1+\beta\big{/}1-\beta)$ and
$\beta=\Omega_{p}/\Delta_{p}^{b}$, we can transform the Hamiltonian
$\mathcal{H}_{\text{fw}}$ to the squeezing picture. We further apply the
rotating-wave approximation
$\Delta_{p}^{a}+\Delta_{p}^{b}\sqrt{1-\beta^{2}}\gg\sinh(r_{p})J$ in the
squeezing picture and neglect the counter-rotating terms. Then, the
Hamiltonian in the frame rotating at frequency of $\Delta_{\text{in}}$ becomes

$\displaystyle\mathcal{H}_{\text{fw}}^{s}/\hbar=$
$\displaystyle\Delta_{a}a_{{}_{\circlearrowright}}^{\dagger}a_{{}_{\circlearrowright}}+i\sqrt{2\kappa_{\text{ex1}}}(a_{\text{in}}a_{{}_{\circlearrowright}}^{\dagger}-a_{\text{in}}^{\dagger}a_{{}_{\circlearrowright}})$
(2)
$\displaystyle+\Delta_{b}^{s}b_{s_{\circlearrowleft}}^{\dagger}b_{s_{\circlearrowleft}}+J_{s}(a_{{}_{\circlearrowright}}^{\dagger}b_{s_{\circlearrowleft}}+b_{s_{\circlearrowleft}}^{\dagger}a_{{}_{\circlearrowright}})\;,$

where $\Delta_{a}=\omega_{a}-\omega_{\text{in}}$,
$\Delta_{b}^{s}=\Delta_{p}^{bs}-\Delta_{\text{in}}$,
$\Delta_{p}^{bs}=\Delta_{p}^{b}\sqrt{1-\beta^{2}}$ and $J_{s}=\cosh(r_{p})J$.
The effective squeezed mode detuning $\Delta_{b}^{s}$ and the effective
coupling rate $J_{s}$ are controlled by the pump field $\Omega_{p}$ and the
corresponding detuning $\Delta_{p}^{b}$. When the ratio $\beta$ approaches
unity, the rate $J_{s}$ between modes $a_{{}_{\circlearrowright}}$ and
$b_{s_{\circlearrowleft}}$ is enhanced exponentially with respect to the bare
coupling rate $J$ (see Supplemental Material [51] for details). Note that the
pump field $\Omega_{p}$ also adds thermalization noise to mode
$b_{s_{\circlearrowleft}}$.

In the squeezing picture, the master equation of the system without a
squeezed-vacuum driving takes the form

$\dot{\rho}_{\text{fw}}=-i[\mathcal{H}_{\text{fw}}^{s},\rho_{\text{fw}}]+\mathcal{L}[L_{a}]\rho_{\text{fw}}+\mathcal{L}[L_{bs}]\rho_{\text{fw}}+\mathcal{\pounds}_{\text{n}}[L_{bs}]\rho_{\text{fw}}\;,$
(3)

where $\rho_{\text{fw}}$ is the system density matrix, the term
$\mathcal{L}[L_{a}]\rho_{\text{fw}}$ ($\mathcal{L}[L_{bs}]\rho_{\text{fw}}$)
with operator $L_{a}=\sqrt{\kappa_{a}}a_{{}_{\circlearrowright}}$
($L_{bs}=\sqrt{\kappa_{b}}b_{s_{\circlearrowleft}}$) describes the decay of
the mode $a_{{}_{\circlearrowright}}$ ($b_{s_{\circlearrowleft}}$) with a rate
$\kappa_{a}$ ($\kappa_{b}$), and $\mathcal{L}[o]\rho=2o\rho
o^{\dagger}-o^{\dagger}o\rho-\rho o^{\dagger}o$. The term
$\mathcal{\pounds}_{\text{n}}[L_{bs}]\rho_{\text{fw}}$ describes the effective
thermalization noise of the mode $b_{s_{\circlearrowleft}}$ resulting from the
_classical coherent_ pump. It is given by
$\mathcal{\pounds}_{\text{n}}[L_{bs}]\rho_{\text{fw}}=N_{p}\mathcal{L}[L_{bs}]\rho_{\text{fw}}+N_{p}\mathcal{L}[L_{bs}^{\dagger}]\rho_{\text{fw}}-M_{p}\mathcal{L}^{\prime}[L_{bs}]\rho_{\text{fw}}-M_{p}^{*}\mathcal{L}^{\prime}[L_{bs}^{\dagger}]\rho_{\text{fw}}$,
where $N_{p}=\sinh^{2}(r_{p})$,
$M_{p}=e^{i\theta_{p}}\cosh(r_{p})\sinh(r_{p})$, and
$\mathcal{L}^{\prime}[o]\rho=2o\rho o-oo\rho-\rho oo$. This noise can limit
the application of the system in the quantum regime.

To eliminate this squeezing-induced noise for achieving single-photon
isolation, we can apply a broadband squeezed-vacuum field $\xi$ with squeezing
parameter $r_{e}$ and reference phase $\theta_{e}$ to drive
$\text{R}_{\text{B}}$ from port 3. When $r_{e}=r_{p}$ and
$\theta_{e}+\theta_{p}=\pm n\pi~{}(n=1,3,5,...)$, the squeezing-induced noise
associated with the term
$\mathcal{\pounds}_{\text{n}}[L_{bs}]\rho_{\text{fw}}$ is cancelled [41, 42,
46, 51]. In doing so, the mode $a_{{}_{\circlearrowright}}$ can couple to the
squeezed mode $b_{s_{\circlearrowleft}}$ coherently without additional noise,
just as a simple linear resonator system. A squeezed-vacuum field with
$\mathrm{T}\mathrm{Hz}$ bandwidth has been realized via optical parametric
amplification [54]. Thus, the bandwidth of the squeezed vacuum is larger than
the linewidth of the resonators in our system. In this case, the squeezed mode
is equivalently coupled to a normal vacuum bath. The decay rate of the
squeezed mode equals to that of the bare mode such that
$\kappa_{bs}=\kappa_{b}$.

For the backward-input case, the Hamiltonian reads

$\displaystyle\mathcal{H}_{\text{bw}}/\hbar=$
$\displaystyle\Delta_{a}a_{{}_{\circlearrowleft}}^{\dagger}a_{{}_{\circlearrowleft}}+i\sqrt{2\kappa_{\text{ex1}}}(a_{\text{in}}a_{{}_{\circlearrowleft}}^{\dagger}-a_{\text{in}}^{\dagger}a_{{}_{\circlearrowleft}})\
$ (4)
$\displaystyle+\Delta_{b}b_{{}_{\circlearrowright}}^{\dagger}b_{{}_{\circlearrowright}}+J(a_{{}_{\circlearrowleft}}^{\dagger}b_{{}_{\circlearrowright}}+b_{{}_{\circlearrowright}}^{\dagger}a_{{}_{\circlearrowleft}})\;,$

where $\Delta_{b}=\omega_{b}-\omega_{\text{in}}$. Comparing Eq. (4) with Eq.
(2), we can see that the intermode detuning and coupling in these two
Hamiltonians can be very different due to the directional quantum squeezing.
The dynamics of the system is governed by the master equation of two coupling
resonators

$\dot{\rho}_{\text{bw}}=-i[\mathcal{H}_{\text{bw}},\rho_{\text{bw}}]+\mathcal{L}[L_{a}]\rho_{\text{bw}}+\mathcal{L}[L_{b}]\rho_{\text{bw}}\;,$
(5)

where $\rho_{\text{bw}}$ is the density matrix of the system,
$L_{a}=\sqrt{\kappa_{a}}a_{{}_{\circlearrowleft}}$, and
$L_{b}=\sqrt{\kappa_{b}}b_{{}_{\circlearrowright}}$. Note that in this case
the squeezed-vacuum field has no influence on the dynamics. Thus, we can
attain strong ONR by parametrically pumping the mode
$b_{{}_{\circlearrowleft}}$.

According to the input-output relation [55], for an input field
$a_{\text{in}}$, we have
$a_{\text{out}}=a_{\text{in}}-\sqrt{2\kappa_{\text{ex1}}}a$ and
$b_{\text{out}}=\sqrt{2\kappa_{\text{ex2}}}b$, where $\kappa_{\text{ex2}}$ is
the external decay rate of $\text{R}_{\text{B}}$. The transmissions are
defined as $T_{12/21}=\langle
a_{\text{out}}^{\dagger}a_{\text{out}}\rangle\big{/}\langle
a_{\text{in}}^{\dagger}a_{\text{in}}\rangle$ and $T_{23}=\langle
b_{\text{out}}^{\dagger}b_{\text{out}}\rangle\big{/}\langle
a_{\text{in}}^{\dagger}a_{\text{in}}\rangle$, where $T_{ij}$ is the
transmission from port $i$ to port $j$, with $i,~{}j=1,~{}2,~{}3$. Numerically
solving Eqs. (3,5) and using $\alpha_{\text{in}}\equiv\langle
a_{\text{in}}\rangle=\sqrt{2\pi P_{\text{in}}/\hbar\omega_{\text{in}}}$ with
the signal power $P_{\text{in}}$, we can obtain the steady-state solutions for
$\langle a_{\text{out}}^{\dagger}a_{\text{out}}\rangle_{\text{ss}}$ and
$\langle b_{\text{out}}^{\dagger}b_{\text{out}}\rangle_{\text{ss}}$, and the
steady-state transmissions.

We can also analytically derive the transmission from the Langevin equations
of motion. To consider the squeezing-induced noise, we truncate the Langevin
equation to second-order nonlinear terms of operators and obtain

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}a_{x}=-\left(i\Delta_{a}+\kappa_{a}\right)a_{x}+\sqrt{2\kappa_{\text{ex1}}}a_{\text{in}}-iJ_{x}b_{x}\;,$
(6)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}b_{x}=-\left(i\Delta_{b}^{x}+\kappa_{b}\right)b_{x}-iJ_{x}a_{x}\;,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}a_{x}^{\dagger}b_{x}=\left(i\Delta_{ab}^{x}-\kappa_{ab}\right)a_{x}^{\dagger}b_{x}+\sqrt{2\kappa_{\text{ex1}}}a_{\text{in}}^{\dagger}b_{x}-iJ_{x}\Xi\;,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}b_{x}^{\dagger}b_{x}=iJ_{x}\left(a_{x}^{\dagger}b_{x}-a_{x}b_{x}^{\dagger}\right)-2\kappa_{b}b_{x}^{\dagger}b_{x}+\Psi_{\text{noise}}\;,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}a_{x}^{\dagger}a_{x}=-iJ_{x}\left(a_{x}^{\dagger}b_{x}-a_{x}b_{x}^{\dagger}\right)+\sqrt{2\kappa_{\text{ex1}}}\left(a_{\text{in}}a_{x}^{\dagger}+a_{\text{in}}^{\dagger}a_{x}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2\kappa_{a}a_{x}^{\dagger}a_{x}\;,$

where the squeezing-induced noise
$\Psi_{\text{noise}}=2\sinh^{2}(r_{p})\kappa_{b}$ is present in the forward-
input case and plays the role of a thermal bath. In the backward-input case,
$\Psi_{\text{noise}}$ is absent. Here, we have used
$\Delta_{ab}^{x}=\Delta_{a}-\Delta_{b}^{x}$,
$\kappa_{ab}=\kappa_{a}+\kappa_{b}$ and
$\Xi=a_{x}^{\dagger}a_{x}-b_{x}^{\dagger}b_{x}$. By solving the Langevin
equation and using the input-output relations, we obtain the steady-state
transmissions:

$\displaystyle T_{12}$
$\displaystyle=\frac{J_{s}^{4}+2\zeta_{x}J_{s}^{2}+\Lambda_{x}}{\mathcal{G}_{x}}+\frac{2\kappa_{\text{ex1}}\mathcal{N}_{\text{noise}}}{|\alpha_{\text{in}}|^{2}}\;,$
(7a) $\displaystyle T_{21}$
$\displaystyle=\frac{J^{4}+2\zeta_{x}J^{2}+\Lambda_{x}}{\mathcal{G}_{x}}\;,$
(7b) $\displaystyle T_{23}$
$\displaystyle=\frac{4\kappa_{\text{ex1}}\kappa_{\text{ex2}}J^{2}}{\mathcal{G}_{x}}\;,$
(7c)

where
$\mathcal{N}_{\text{noise}}=\kappa_{b}(\kappa_{a}+\kappa_{b})\sinh^{2}(r_{p})J_{s}^{2}/\mathcal{Q}_{s}$
is the number of noise-related photons,
$\mathcal{Q}_{s}=J_{s}^{2}(\kappa_{a}+\kappa_{b})^{2}+\kappa_{a}\kappa_{b}[(\kappa_{a}+\kappa_{b})^{2}+{\Delta_{ab}^{s}}^{2}]$,
$\mathcal{G}_{x}=J_{x}^{4}+2J_{x}^{2}(\kappa_{a}\kappa_{b}-\Delta_{a}\Delta_{b}^{x})+(\kappa_{a}^{2}+\Delta_{a}^{2})(\kappa_{b}^{2}+{\Delta_{b}^{x}}^{2})$,
$\zeta_{x}=\kappa_{a}\kappa_{b}-2\kappa_{b}\kappa_{\text{ex1}}-\Delta_{a}\Delta_{b}^{x}$,
and
$\Lambda_{x}=[(\kappa_{a}-2\kappa_{\text{ex1}})^{2}+\Delta_{a}^{2}](\kappa_{b}^{2}+{\Delta_{b}^{x}}^{2})$.
In the calculations, we need to replace $a_{x}$ with
$a_{{}_{\circlearrowright}}$, $b_{x}$ with $b_{s_{\circlearrowleft}}$, $J_{x}$
with $J_{s}$, and $\Delta_{b}^{x}$ with $\Delta_{b}^{s}$ ($a_{x}$ with
$a_{{}_{\circlearrowleft}}$, $b_{x}$ with $b_{{}_{\circlearrowright}}$,
$J_{x}$ with $J$, and $\Delta_{b}^{x}$ with $\Delta_{b}$) for the forward-
input (backward-input) case (see Supplemental Material [51]).

By applying the squeezed-vacuum field to cancel the noise
$\Psi_{\text{noise}}$, yielding $\Psi_{\text{noise}}=0$ and
$\mathcal{N}_{\text{noise}}=0$. The steady-state transmissions become

$T_{12}^{\text{sv}}=(J_{s}^{4}+2\zeta_{s}J_{s}^{2}+\Lambda_{s})\big{/}\mathcal{G}_{s}\;,\quad
T_{21}^{\text{sv}}=T_{21}\;,\quad T_{23}^{\text{sv}}=T_{23}\;.$ (8)

The noise-free transmission $T_{12}^{\text{sv}}$ is the limitation of $T_{12}$
for a classical large input $\alpha_{\text{in}}$ and also valid for single-
photon pulses. Thus, we can achieve ONR in both the classical and quantum
regimes.

Below, we will show two different mechanisms to achieve a strong ONR
dependence on the bare mode coupling rate $J$. When $J<\kappa_{a},\kappa_{b}$,
the two resonators originally have no NMS. We use the pump field to induce the
NMS between $a_{{}_{\circlearrowright}}$ and $b_{s_{\circlearrowleft}}$,
namely the NMS scenario. For $J\gg\kappa_{a},\kappa_{b}$ resulting in the NMS
between the bare modes, we use the pump field to significantly shift the
resonance frequency of the mode $b_{s_{\circlearrowleft}}$. We call this
mechanism the mode resonance shift (MRS) scenario.

Figure 2:  (a) and (c) Steady-state transmission and isolation (inset) versus
$\Delta_{a}$. (b) and (d) Isolation and insertion loss versus $r_{p}$. (a) and
(b) for the NMS scenario ($J/\kappa_{a}=0.99$), (c) and (d) for the MRS
scenario ($J/\kappa_{a}=2.8$). Curves and circles are for analytical and
numerical transmissions of $T_{12}^{(\text{sv})}$ (blue),
$T_{21}^{(\text{sv})}$ (red) and $T_{23}^{(\text{sv})}$ (green), respectively.
Solid (dashed) curves are for analytical transmissions $T_{ij}^{\text{sv}}$
($T_{ij}$) with (without) the squeezed-vacuum field. Circles show the
corresponding numerical transmissions $T_{ij}$. Other parameters are:
$\Delta_{a}=\Delta_{b}$, $\kappa_{a}=\kappa_{b}$,
$\kappa_{\text{ex1,2}}/\kappa_{a}=0.99$; in (a)
$\Delta_{p}^{b}/\kappa_{a}=10.3$, $\Omega_{p}/\kappa_{a}=10$, yielding
$r_{p}\sim 1.05$; in (c) $\Delta_{p}^{b}/\kappa_{a}=15$,
$\Omega_{p}/\kappa_{a}=13$, yielding $r_{p}\sim 0.66$; in (b) $\Delta_{a}=0$
and $\Delta_{p}^{b}=10\sinh(r_{p})$; in (d) $\Delta_{a}/\kappa_{a}=2.62$ and
$\Delta_{p}^{b}=30\sinh(r_{p})$.

Our system can realize an optical diode with transmission $T_{12}\gg T_{21}$
even with only the _classical coherent_ pump $\Omega_{p}$. For a weak input
signal, e.g. $\alpha_{\text{in}}/\sqrt{\kappa_{a}}=0.6$, the transmission
$T_{12}$ (blue dashed curves) can be larger than unity, because the pump-
induced noise can thermalize the squeezed mode. As the intensity of the
incident signal increases to $\alpha_{\text{in}}/\sqrt{\kappa_{a}}>3$,
corresponding to a weak signal including $9$ photons within the resonator
decay time, the input signal overwhelms the pump-induced noise. In this case,
the transmission $T_{12}$ in the classical regime approximates the noise-free
transmission $T_{12}^{\text{sv}}$ (blue solid curves) in the quantum regime.
The analytical transmissions are in excellent agreement with the numerical
results, see Figs. 2(a,c). Below, we focus on the classical case with a large
signal input or the noise-free transmission when the squeezed-vacuum field is
present. In the NMS scenario, we choose $J=0.99\kappa_{a}$ for the critical
coupling condition, such that the transmission $T_{21}$ vanishes in the
backward-input case. In comparison, in the forward-input case, the squeezing
interaction enhances the coupling strength $J_{s}$ to much larger than the
system decay rates and leads to a large NMS, see Fig. 2(a). Therefore, we
obtain $T_{12}^{\text{sv}}\approx 83.1\%$ and $T_{21}^{\text{sv}}\approx 0$ at
$\Delta_{a}=0$, corresponding to an isolation ratio
$\eta=10\log(T_{12}^{\text{sv}}/T_{21}^{\text{sv}})\approx
85.1~{}\mathrm{d}\mathrm{B}$, insertion loss
${L}=-10\log(T_{12}^{\text{sv}})\approx 0.80~{}\mathrm{d}\mathrm{B}$. The
bandwidth for an isolation ratio larger than $20~{}\mathrm{d}\mathrm{B}$ is
about $0.86\kappa_{a}$, see the inset of Fig. 2(a). In the MRS scenario, we
take $J=2.8\kappa_{a}$ for example. The transmission spectrum splits in the
backward-input case. In the forward-input case, the resonance frequency of the
squeezed mode is shifted and thus detuned from the mode
$a_{{}_{\circlearrowright}}$. The transmission spectrum is shifted with
respect to the backward-input transmission, see Fig. 2(c). As a result, we
attain $T_{12}^{\text{sv}}\approx 92.8\%$ and $T_{21}^{\text{sv}}\approx 0$ at
$\Delta_{a}/\kappa_{a}=2.62$, corresponding to $\eta\approx
40.3~{}\mathrm{d}\mathrm{B}$, ${L}\approx 0.32~{}\mathrm{d}\mathrm{B}$. The
bandwidth for $\eta\geqslant 20~{}\mathrm{d}\mathrm{B}$ is about
$0.21\kappa_{a}$, see the inset of Fig. 2(c). The isolation ratio and the
insertion loss improve with $r_{p}$ and reach stable values when $r_{p}>0.6$,
see Figs. 2(b,d).

Similar to a commercial circulator, our proposed system can also function as a
quasi-circulator with two inputs and three outputs, allowing photon flow along
the direction $1\rightarrow 2\rightarrow 3$ [56], see Figs. 2(a,c). Here, we
focus on the fidelity and the insertion loss. To evaluate the quasi-circulator
performance, we calculate the average fidelity as
$\mathcal{F}=\text{Tr}\left[\tilde{T}{T^{\text{id}}}^{T}\right]\bigg{/}\text{Tr}\left[T^{\text{id}}{T^{\text{id}}}^{T}\right]$
[17, 32], where $T^{\text{id}}$ is the transmission matrix for an ideal three-
port quasi-circulator [56], and $\tilde{T}=T_{ij}/\Upsilon_{i}$, with
$\Upsilon_{i}=\textstyle\sum_{j}T_{ij}$. We define the average insertion loss
as $\tilde{{L}}=-10\log[(T_{12}+T_{23})/2]$. The transmissions $T_{12}$ and
$T_{21}$ are the same as the diode. For an input
$\alpha_{\text{in}}/\sqrt{\kappa_{a}}>3$, the pump-induced noise is
negligible. In the NMS scenario, we achieve $T_{23}\approx 98.0\%$ at
$\Delta_{a}=0$, corresponding to $\tilde{{L}}\approx
0.43~{}\mathrm{d}\mathrm{B}$. We also obtain similar performance in the MRS
scenario with $T_{23}\approx 98.0\%$ at $\Delta_{a}/\kappa_{a}=2.62$,
corresponding to $\tilde{{L}}\approx 0.20~{}\mathrm{d}\mathrm{B}$. In both
scenarios, we have $\mathcal{F}\approx 1$.

Our system can work as a single-photon quasi-circulator when the squeezing-
vacuum field is applied to cancel the noise term. We evaluate its performance
for a single-photon wavepacket input to ports $1$ and $2$ simultaneously by
solving a quantum cascaded system [57, 58, 59] (also Supplemental Material
[51]). We consider a Gaussian-like single-photon pulse with duration
$2\pi\times 6\kappa_{a}^{-1}$. In comparison with the case without the
squeezed-vacuum field, the performance of our quasi-circulator only slightly
decreases. The fidelity is still very high, $\mathcal{F}\geq 98.7\%$. The
insertion loss remains unchanged due to the large bandwidth in the NMS
scenario. In comparison, it reduces to $\tilde{{L}}\approx
0.26~{}\mathrm{d}\mathrm{B}$ at $\Delta_{a}/\kappa_{a}=2.62$ in the MRS
scenario. Therefore, our diode and quasi-circulator can work in both the
classical and quantum regimes.

When the pump laser with power $P_{p}$ is present, corresponding to “ON”
(absent, corresponding to “OFF”), our device can switch on and off the
transmission of a stronger signal laser from $T_{12}^{\text{on}}$ to
$T_{12}^{\text{off}}$ according to Eq. 7(a), implying an all-optical
nonreciprocal transistor. As the crucial role of electronic transistor in
electric computers, optical transistors are essential for optical information
processing [60, 61, 62, 63]. We define the gain of the transistor as
$G=\frac{P_{\text{in}}}{P_{p}}\Delta T$, with $\Delta
T=T_{12}^{\text{on}}-T_{12}^{\text{off}}$ [60]. The gain increases linearly
with the signal power. When we fix the pump strength, e.g.
$\Omega_{p}/\kappa_{a}=10$, in the NMS scenario, the gain of the transistor
can reach $G>1$ ($G>100$) when $|\alpha_{\text{in}}|^{2}/\kappa_{a}>6.1\times
10^{7}$ ($|\alpha_{\text{in}}|^{2}/\kappa_{a}>6.1\times 10^{9}$). In the MRS
scenario, we can obtain the same gain $G>1$ ($G>100$) by taking
$\Omega_{p}/\kappa_{a}=13$ and applying a slightly larger pump power
$|\alpha_{\text{in}}|^{2}/\kappa_{a}>9.2\times 10^{7}$
($|\alpha_{\text{in}}|^{2}/\kappa_{a}>9.2\times 10^{9}$). Unlike the forward-
input case, the transmission $T_{21}$ in the backward-input case is
independent of the pump power and always vanishing small. Clearly, our optical
transistor is nonreciprocal.

Figure 3: Gain of the transistor versus detuning $\Delta_{a}$ and
$P_{\text{in}}$. (a) The NMS scenario with $J/\kappa_{a}=0.99$,
$\Delta_{p}^{b}/\kappa_{a}=10.3$, $\Omega_{p}/\kappa_{a}=10$. (b) The MRS
scenario with $J/\kappa_{a}=2.8$, $\Delta_{p}^{b}/\kappa_{a}=15$,
$\Omega_{p}/\kappa_{a}=13$. Other parameters are: $\Delta_{a}=\Delta_{b}$,
$\kappa_{a}=\kappa_{b}$, $\kappa_{\text{ex1,2}}/\kappa_{a}=0.99$,
$g=10^{-3}\kappa_{a}$.

Lithium-niobate-based microring resonators provide an excellent platform for
our proposal, thanks to their large $\chi^{(2)}$ [64, 65, 66] and high optical
quality factors up to $Q\sim 10^{7}$ [67, 68]. Assuming an experimentally
available intrinsic quality factor $Q_{i}=8\times 10^{6}$, the resonance
frequency of the signal field
$\omega_{a/b}/2\pi=193.4~{}\mathrm{T}\mathrm{Hz}$ and the pump field frequency
$\omega_{p}=2\omega_{a}$ for the resonators, we obtain the intrinsic loss rate
$\kappa_{i}/2\pi=24.2~{}\mathrm{M}\mathrm{Hz}$. By choosing an experimentally
available gap [69], we can select feasible values for the external decay rates
and the coupling rate: $\kappa_{\text{ex1}}=\kappa_{\text{ex2}}\approx
2\pi\times 2.40~{}\mathrm{G}\mathrm{Hz}$, $\kappa_{a}=\kappa_{b}\approx
2\pi\times 2.42~{}\mathrm{G}\mathrm{Hz}$ and $J=0.99\kappa_{a}\approx
2\pi\times 2.40~{}\mathrm{G}\mathrm{Hz}$ in the NMS scenario (or
$J=2.8\kappa_{a}\approx 2\pi\times 6.78~{}\mathrm{G}\mathrm{Hz}$ in the MRS
scenario). Hence, the nonreciprocal bandwidth for $\eta\geq
20~{}\mathrm{d}\mathrm{B}$ can reach $0.86\kappa_{a}/2\pi\approx
2.08~{}\mathrm{G}\mathrm{Hz}$ around $\Delta_{a}=0$ (or
$0.21\kappa_{a}/2\pi\approx 0.51~{}\mathrm{G}\mathrm{Hz}$ around
$\Delta_{a}=2.62\kappa_{a}$). The power of a pump field is given by
$P_{p}={\hbar\omega_{p}\kappa_{p}^{2}\Omega_{p}^{2}}/{16\pi
g^{2}\kappa_{\text{ex2}}^{p}}$ [51]. The rate of
$g/2\pi=2.35~{}\mathrm{M}\mathrm{Hz}$ is available for Lithium-niobate-based
microresonators [70]. Thus, the pump power $P_{p}\approx
16.6~{}\mathrm{m}\mathrm{W}$ (or $28.0~{}\mathrm{m}\mathrm{W}$) yields to
$\Omega_{p}/\kappa_{a}=10$ (or $13$). At the detuning
$\Delta_{a}/\kappa_{a}=0$ (or $2.62$), we can induce an optical transistor
with $G>1$ for a signal power $P_{\text{in}}\approx
18.9~{}\mathrm{m}\mathrm{W}$ in the NMS scenario (or
$28.5~{}\mathrm{m}\mathrm{W}$ in the MRS scenario).

We have proposed a chip-compatible magnetic-free system to realize an optical
diode, a quasi-circulator and a nonreciprocal optical transistor. The optical
diode and circulator can work _in the classical regime under a coherent pump
and also in the quantum regime when a squeezed-vacuum field is applied_. Our
method only needs two-mode phase matching in one resonator. It greatly
simplifies its experimental implementation, in comparison with a non-
degenerate three-mode matching counterpart [39, 40]. In particular, our work
proposed the nonreciprocal optical transistor switching a strong signal with a
weak control field. Such unconventional transistor cannot be realized in the
configuration of Ref. [39] because a strong signal will cause NMS for the pump
field.

## Acknowledgements

This work was supported by the National Key R&D Program of China (Grants No.
2017YFA0303703, No. 2019YFA0308700), the National Natural Science Foundation
of China (Grant Nos. 11874212, 11890704, 11690031), the Fundamental Research
Funds for the Central Universities (Grant No. 021314380095), the Program for
Innovative Talents and Entrepreneurs in Jiangsu, and the Excellent Research
Program of Nanjing University (Grant No. ZYJH002). F.N. is supported by the
Nippon Telegraph and Telephone Corporation (NTT) Research, the Japan Science
and Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP),
the Moonshot R&D Grant Number JPMJMS2061, and the Centers of Research
Excellence in Science and Technology (CREST) Grant No. JPMJCR1676], the Japan
Society for the Promotion of Science (JSPS) [via the Grants-in-Aid for
Scientific Research (KAKENHI) Grant No. JP20H00134 and the JSPS-RFBR Grant No.
JPJSBP120194828], the Army Research Office (ARO) (Grant No. W911NF-18-1-0358),
the Asian Office of Aerospace Research and Development (AOARD) (via Grant No.
FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via
Grant No. FQXi-IAF19-06. We thank the High Performance Computing Center of
Nanjing University for doing the numerical calculations on its blade cluster
system.

## References

  * Fan  _et al._ [2012] L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, An all-silicon passive optical diode, Science 335, 447 (2012).
  * Peng  _et al._ [2014] B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Parity-time-symmetric whispering-gallery microcavities, Nat. Phys. 10, 394 (2014).
  * Chang  _et al._ [2014] L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators, Nat. Photonics 8, 524 (2014).
  * Shi  _et al._ [2015] Y. Shi, Z. Yu, and S. Fan, Limitations of nonlinear optical isolators due to dynamic reciprocity, Nat. Photonics 9, 388 (2015).
  * Hua  _et al._ [2016] S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, Demonstration of a chip-based optical isolator with parametric amplification, Nat. Commun. 7, 13657 (2016).
  * He  _et al._ [2018] B. He, L. Yang, X. Jiang, and M. Xiao, Transmission nonreciprocity in a mutually coupled circulating structure, Phys. Rev. Lett. 120, 203904 (2018).
  * Bino  _et al._ [2018] L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect, Optica 5, 279 (2018).
  * Yang  _et al._ [2020] K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, Inverse-designed non-reciprocal pulse router for chip-based lidar, Nat. Photonics 14, 369 (2020).
  * Tang  _et al._ [2021] L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity, Photon. Res. 9, 1218 (2021).
  * Lira  _et al._ [2012] H. Lira, Z. Yu, S. Fan, and M. Lipson, Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip, Phys. Rev. Lett. 109, 033901 (2012).
  * Estep  _et al._ [2014] N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops, Nat. Phys. 10, 923 (2014).
  * Sounas and Alù [2017] D. L. Sounas and A. Alù, Non-reciprocal photonics based on time modulation, Nat. Photonics 11, 774 (2017).
  * Xia  _et al._ [2014] K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling, Phys. Rev. A 90, 043802 (2014).
  * Bliokh  _et al._ [2015a] K. Y. Bliokh, D. Smirnova, and F. Nori, Quantum spin Hall effect of light, Science 348, 1448 (2015a).
  * Bliokh and Nori [2015] K. Y. Bliokh and F. Nori, Transverse and longitudinal angular momenta of light, Phys. Rep. 592, 1 (2015).
  * Bliokh  _et al._ [2015b] K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Spin-orbit interactions of light, Nat. Photonics 9, 796 (2015b).
  * Scheucher  _et al._ [2016] M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, Quantum optical circulator controlled by a single chirally coupled atom, Science 354, 1577 (2016).
  * Lodahl  _et al._ [2017] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Chiral quantum optics, Nature (London) 541, 473 (2017).
  * Bliokh  _et al._ [2019] K. Y. Bliokh, D. Leykam, M. Lein, and F. Nori, Topological non-Hermitian origin of surface Maxwell waves, Nat. Commun. 10, 580 (2019).
  * Tang  _et al._ [2019] L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, On-chip chiral single-photon interface: Isolation and unidirectional emission, Phys. Rev. A 99, 043833 (2019).
  * Guddala  _et al._ [2021] S. Guddala, Y. Kawaguchi, F. Komissarenko, S. Kiriushechkina, A. Vakulenko, K. Chen, A. Alù, V. M. Menon, and A. B. Khanikaev, All-optical nonreciprocity due to valley polarization pumping in transition metal dichalcogenides, Nat. Commun. 12, 3746 (2021).
  * Manipatruni  _et al._ [2009] S. Manipatruni, J. T. Robinson, and M. Lipson, Optical nonreciprocity in optomechanical structures, Phys. Rev. Lett. 102, 213903 (2009).
  * Shen  _et al._ [2016] Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, Experimental realization of optomechanically induced non-reciprocity, Nat. Photonics 10, 657 (2016).
  * Shen  _et al._ [2018] Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, Reconfigurable optomechanical circulator and directional amplifier, Nat. Commun. 9, 1797 (2018).
  * Wang  _et al._ [2013] D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, Optical diode made from a moving photonic crystal, Phys. Rev. Lett. 110, 093901 (2013).
  * Horsley  _et al._ [2013] S. A. R. Horsley, J.-H. Wu, M. Artoni, and G. C. La Rocca, Optical nonreciprocity of cold atom Bragg mirrors in motion, Phys. Rev. Lett. 110, 223602 (2013).
  * Wu  _et al._ [2014] J.-H. Wu, M. Artoni, and G. C. La Rocca, Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices, Phys. Rev. Lett. 113, 123004 (2014).
  * Lu  _et al._ [2021] X. Lu, W. Cao, W. Yi, H. Shen, and Y. Xiao, Nonreciprocity and quantum correlations of light transport in hot atoms via reservoir engineering, Phys. Rev. Lett. 126, 223603 (2021).
  * Maayani  _et al._ [2018] S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, Flying couplers above spinning resonators generate irreversible refraction, Nature (London) 558, 569 (2018).
  * Huang  _et al._ [2018] R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, Nonreciprocal photon blockade, Phys. Rev. Lett. 121, 153601 (2018).
  * Jiao  _et al._ [2020] Y.-F. Jiao, S.-D. Zhang, Y.-L. Zhang, A. Miranowicz, L.-M. Kuang, and H. Jing, Nonreciprocal optomechanical entanglement against backscattering losses, Phys. Rev. Lett. 125, 143605 (2020).
  * Xia  _et al._ [2018] K. Xia, F. Nori, and M. Xiao, Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity, Phys. Rev. Lett. 121, 203602 (2018).
  * Zhang  _et al._ [2018] S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, Thermal-motion-induced non-reciprocal quantum optical system, Nat. Photonics 12, 744 (2018).
  * Hu  _et al._ [2019] Y. Hu, S. Zhang, Y. Qi, G. Lin, Y. Niu, and S. Gong, Multiwavelength magnetic-free optical isolator by optical pumping in warm atoms, Phys. Rev. Applied 12, 054004 (2019).
  * Yang  _et al._ [2019] P. Yang, X. Xia, H. He, S. Li, X. Han, P. Zhang, G. Li, P. Zhang, J. Xu, Y. Yang, and T. Zhang, Realization of nonlinear optical nonreciprocity on a few-photon level based on atoms strongly coupled to an asymmetric cavity, Phys. Rev. Lett. 123, 233604 (2019).
  * Liang  _et al._ [2020] C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, Collision-induced broadband optical nonreciprocity, Phys. Rev. Lett. 125, 123901 (2020).
  * Dong  _et al._ [2021] M.-X. Dong, K.-Y. Xia, W.-H. Zhang, Y.-C. Yu, Y.-H. Ye, E.-Z. Li, L. Zeng, D.-S. Ding, B.-S. Shi, G.-C. Guo, and F. Nori, All-optical reversible single-photon isolation at room temperature, Sci. Adv. 7, eabe8924 (2021).
  * Hu  _et al._ [2021] Y. Hu, Y. Qi, Y. You, S. Zhang, G. Lin, X. Li, J. Gong, S. Gong, and Y. Niu, Passive nonlinear optical isolators bypassing dynamic reciprocity, Phys. Rev. Applied 16, 014046 (2021).
  * Guo  _et al._ [2016] X. Guo, C.-L. Zou, H. Jung, and H. X. Tang, On-chip strong coupling and efficient frequency conversion between telecom and visible optical modes, Phys. Rev. Lett. 117, 123902 (2016).
  * Wang  _et al._ [2021] J.-Q. Wang, Y.-H. Yang, M. Li, X.-X. Hu, J. B. Surya, X.-B. Xu, C.-H. Dong, G.-C. Guo, H. X. Tang, and C.-L. Zou, Efficient frequency conversion in a degenerate ${\chi}^{(2)}$ microresonator, Phys. Rev. Lett. 126, 133601 (2021).
  * Lü  _et al._ [2015] X.-Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, Squeezed optomechanics with phase-matched amplification and dissipation, Phys. Rev. Lett. 114, 093602 (2015).
  * Qin  _et al._ [2018] W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lü, J. Q. You, and F. Nori, Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification, Phys. Rev. Lett. 120, 093601 (2018).
  * Leroux  _et al._ [2018] C. Leroux, L. C. G. Govia, and A. A. Clerk, Enhancing cavity quantum electrodynamics via antisqueezing: Synthetic ultrastrong coupling, Phys. Rev. Lett. 120, 093602 (2018).
  * Ge  _et al._ [2019] W. Ge, B. C. Sawyer, J. W. Britton, K. Jacobs, J. J. Bollinger, and M. Foss-Feig, Trapped ion quantum information processing with squeezed phonons, Phys. Rev. Lett. 122, 030501 (2019).
  * Zhao  _et al._ [2019] W. Zhao, S.-D. Zhang, A. Miranowicz, and H. Jing, Weak-force sensing with squeezed optomechanics, Sci. China-Phys. Mech. Astron. 63, 224211 (2019).
  * Qin  _et al._ [2019] W. Qin, V. Macrì, A. Miranowicz, S. Savasta, and F. Nori, Emission of photon pairs by mechanical stimulation of the squeezed vacuum, Phys. Rev. A 100, 062501 (2019).
  * Zhu  _et al._ [2020] C. J. Zhu, L. L. Ping, Y. P. Yang, and G. S. Agarwal, Squeezed light induced symmetry breaking superradiant phase transition, Phys. Rev. Lett. 124, 073602 (2020).
  * Chen  _et al._ [2021] Y.-H. Chen, W. Qin, X. Wang, A. Miranowicz, and F. Nori, Shortcuts to adiabaticity for the quantum rabi model: Efficient generation of giant entangled cat states via parametric amplification, Phys. Rev. Lett. 126, 023602 (2021).
  * Burd  _et al._ [2021] S. C. Burd, R. Srinivas, H. M. Knaack, W. Ge, A. C. Wilson, D. J. Wineland, D. Leibfried, J. J. Bollinger, D. T. C. Allcock, and D. H. Slichter, Quantum amplification of boson-mediated interactions, Nat. Phys. 17, 898 (2021).
  * Qin  _et al._ [2021] W. Qin, A. Miranowicz, H. Jing, and F. Nori, Generating long-lived macroscopically distinct superposition states in atomic ensembles, Phys. Rev. Lett. 127, 093602 (2021).
  * [51] See the Supplemental Material for more details .
  * Scully and Zubairy [1997] M. O. Scully and M. S. Zubairy,  _Quantum Optics_ (Cambridge University Press, Cambridge, England, 1997).
  * Agarwal [2012] G. S. Agarwal,  _Quantum Optics_ (Cambridge University Press, Cambridge, England, 2012).
  * Kashiwazaki  _et al._ [2020] T. Kashiwazaki, N. Takanashi, T. Yamashima, T. Kazama, K. Enbutsu, R. Kasahara, T. Umeki, and A. Furusawa, Continuous-wave 6-dB-squeezed light with 2.5-THz-bandwidth from single-mode PPLN waveguide, APL Photonics 5, 036104 (2020).
  * Gardiner and Collett [1985] C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation, Phys. Rev. A 31, 3761 (1985).
  * Kutsaev  _et al._ [2021] S. V. Kutsaev, A. Krasnok, S. N. Romanenko, A. Y. Smirnov, K. Taletski, and V. P. Yakovlev, Up-and-coming advances in optical and microwave nonreciprocity: From classical to quantum realm, Adv. Photonics Res. 2, 2000104 (2021).
  * Gardiner [1993] C. W. Gardiner, Driving a quantum system with the output field from another driven quantum system, Phys. Rev. Lett. 70, 2269 (1993).
  * Carmichael [1993] H. J. Carmichael, Quantum trajectory theory for cascaded open systems, Phys. Rev. Lett. 70, 2273 (1993).
  * Stannigel  _et al._ [2012] K. Stannigel, P. Rabl, and P. Zoller, Driven-dissipative preparation of entangled states in cascaded quantum-optical networks, New J. Phys. 14, 063014 (2012).
  * Sun  _et al._ [2018] S. Sun, H. Kim, Z. Luo, G. S. Solomon, and E. Waks, A single-photon switch and transistor enabled by a solid-state quantum memory, Science 361, 57 (2018).
  * Chen  _et al._ [2013] W. Chen, K. M. Beck, R. Bücker, M. Gullans, M. D. Lukin, H. Tanji-Suzuki, and V. Vuletić, All-optical switch and transistor gated by one stored photon, Science 341, 768 (2013).
  * Hwang  _et al._ [2009] J. Hwang, M. Pototschnig, R. Lettow, G. Zumofen, A. Renn, S. Götzinger, and V. Sandoghdar, A single-molecule optical transistor, Nature (London) 460, 76 (2009).
  * Ballarini  _et al._ [2013] D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdré, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, and D. Sanvitto, All-optical polariton transistor, Nat. Commun. 4, 1778 (2013).
  * Wang  _et al._ [2017] C. Wang, X. Xiong, N. Andrade, V. Venkataraman, X.-F. Ren, G.-C. Guo, and M. Lončar, Second harmonic generation in nano-structured thin-film lithium niobate waveguides, Opt. Express 25, 6963 (2017).
  * Wang  _et al._ [2020] M. Wang, N. Yao, R. Wu, Z. Fang, S. Lv, J. Zhang, J. Lin, W. Fang, and Y. Cheng, Strong nonlinear optics in on-chip coupled lithium niobate microdisk photonic molecules, New J. Phys. 22, 073030 (2020).
  * Ma  _et al._ [2020] Z. Ma, J.-Y. Chen, Z. Li, C. Tang, Y. M. Sua, H. Fan, and Y.-P. Huang, Ultrabright quantum photon sources on chip, Phys. Rev. Lett. 125, 263602 (2020).
  * Zhang  _et al._ [2017] M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and M. Lončar, Monolithic ultra-high-Q lithium niobate microring resonator, Optica 4, 1536 (2017).
  * Desiatov  _et al._ [2019] B. Desiatov, A. Shams-Ansari, M. Zhang, C. Wang, and M. Lončar, Ultra-low-loss integrated visible photonics using thin-film lithium niobate, Optica 6, 380 (2019).
  * Tian  _et al._ [2021] Z. Tian, P. Zhang, and X.-W. Chen, Static hybrid quantum nodes: Toward perfect state transfer on a photonic chip, Phys. Rev. Applied 15, 054043 (2021).
  * Lu  _et al._ [2020] J. Lu, M. Li, C.-L. Zou, A. A. Sayem, and H. X. Tang, Toward 1% single-photon anharmonicity with periodically poled lithium niobate microring resonators, Optica 7, 1654 (2020).