# Optomechanical second-order sidebands and group delays in a spinning
resonator with parametric amplifier and non-Markovian effects

Wei Zhang1 and H. Z. Shen1,2,111 Corresponding author<EMAIL_ADDRESS>1Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China
2Center for Advanced Optoelectronic Functional Materials Research, and Key
Laboratory for UV Light-Emitting Materials and Technology of Ministry of
Education, Northeast Normal University, Changchun 130024, China

###### Abstract

We investigate the generation of the frequency components at the second-order
sidebands based on a spinning resonator containing a degenerate optical
parametric amplifier (OPA). We show an OPA driven by different pumping
frequencies inside a cavity can enhance and modulate the amplitude of the
second-order sideband with different influences. We find that both the second-
order sideband amplitude and its associated group delay sensitively depend on
the nonlinear gain of the OPA, the phase of the field driving the OPA, the
rotation speed of the resonator, and the incident direction of the input
fields. Tuning the pumping frequency of the OPA can remain the localization of
the maximum value of the sideband efficiency and nonreciprocal behavior due to
the optical Sagnac effect, which also can adjust the linewidth of the
suppressive window of the second-order sideband. Furthermore, we extend the
study of second-order sideband to the non-Markovian bath which consists of a
collection of infinite oscillators (bosonic photonic modes). We illustrate the
second-order sidebands in a spinning resonator exhibit a transition from the
non-Markovian to Markovian regime by controlling environmental spectral width.
We also study the influences of the decay from the non-Markovian environment
coupling to an external reservoir on the efficiency of second-order upper
sidebands. This indicates a promising new way to enhance or steer
optomechanically induced transparency devices in nonlinear optical cavities
and provides potential applications for precision measurement, optical
communications, and quantum sensing.

##  I Introduction

In recent years, quantities of attention have been paid to the field of
optomechanics Aspelmeyer861391 ; Aspelmeyer6529 ; Kippenberg3211172 ;
Marquardt240 ; Sainadh92033824 , in which different considerable phenomena
have been met. There are different applications such as cooling of a
mechanical resonator Metzger4321002 ; Gigan44467 ; Arcizet44471 ;
Schliesser5509 ; Meystre525215 , gravitational wave detection Caves4575 ;
Abramovici256325 ; Braginsky293228 , optical bistability Nejad562816 ;
Sarma331335 ; Shahidani311087 , optomechanical mass sensors Jiang2213773 ,
quantum measurement Thompson45272 , and detection of weak microwave signals
Bagci50781 ; Andrews10321 ; Nejad97053839  in merged quantum mechanical
systems with nano and micro mechanics. The recent advance in connection with
the present study closely is optomechanically induced transparency (OMIT)
Weis3301520 ; Safavi47269 ; Jia91043843 ; Jing59663 ; Wang90023817 . In OMIT,
the intense red-detuned optical control field produces anti-Stokes scattering,
which alters the optical response of the optomechanical cavity, making it
transparent in a narrow bandwidth around the cavity resonance for a probe beam
Karuza88013804 . As an analog of electromagnetically induced transparency
Fleischhauer77633 ; Agarwal81041803 , OMIT plays an essential role in optical
storage and optical telecommunication Chang13023003 ; Fiore107133601 ;
Zhou9179 ; Hill31196 . In the last several years, the main progress has
concentrated on the linearization of the optomechanical interaction, where we
properly explain OMIT by linearizing the optomechanical interaction in the
case of ignoring the intrinsic nonlinear nature of the optomechanical coupling
Agarwal81041803 ; Huang83043826 . In recent years, nonlinear optical
interactions in materials can increase the photons circulating in
microcavities, such as parametric amplification and optical Kerr effect
Xiong58050302 ; Bartolo94033841 ; Zhou421289 ; Ilchenko92043903 , which has
emerged as an important new frontier in cavity optomechanics. In the classical
mechanism, nonlinear optomechanical interaction brings about unconventional
photon blockade Rabl107063601 ; Flayac96053810 ; Lemonde90063824 ,
optomechanical chaos Marino87052906 , and sideband generation Xiong86013815 .

Nonreciprocal transmission plays a very important role in the process of
quantum information Bi5758 ; Aleahmad72129 ; AbdelMalek528560 ; Bernier8604
due to the characteristics of unidirectional transmission. The nonreciprocal
transmission of the optical signal allows the flow of light from one side but
blocks it from the other, which resembles the traditional semiconductor p-n
junction. Recently, OMIT has been demonstrated in a rotating optomechanical
system with a whispering-gallery-mode (WGM) microresonator Schliesser10095015
; H5367 ; Jiang3371 . The experiment Maayani558569  shows that optical
nonreciprocal devices can be achieved by spinning an optomechanical resonator.
In such a spinning resonator, due to the Sagnac effect, the frequencies of the
clockwise and counterclockwise modes experience Sagnac-Fizeau shifts.
Additionally, it also suggests a new scheme to achieve optical nonreciprocity
that the optical sidebands strongly rely on the rotary direction of the
resonator, which is different from the nonlinearity-based schemes demonstrated
Shen10657 ; Ruesink713662 ; Fang7465 ; Cao118033901 ; Li102033526 . The
spinning resonator systems have developed rapidly, including nanoparticles
sensing Jing51424 , mass sensing Chen14082005 , nonreciprocal photon blockades
Huang121153601 ; Li7630 , nonreciprocal phonon lasers Jiang10064037 ,
unidirectional signal amplification Peng5332000405 , breaking anti-PT symmetry
Zhang207594 , and optical solitons Li103053522 .

It has been shown that combining nonlinear optics and optomechanics has
resulted in many kinds of physical phenomena to enhance quantum effects
Otey93033835 ; Coillet9828 . An optical parametric amplifier (OPA) inside the
optomechanical cavity, which is pumped by an external laser, can directly lead
to optical amplification and modulate the optomechanical coupling in a way
analogous to periodic cavity driving Adamyan92053818 ; Hu100043824 ;
Lu114093602 . The OPA is able to generate pairs of down-converted photons,
which shows nearly perfect single or dual squeezing. Therefore, the OPA can
modify the dynamical instabilities and nonlinear dynamics of the system
Mi67115 ; Hu7124 ; Xuereb86013809 . Numerous applications have been studied
owing to these features, such as the realization of strong mechanical
squeezing Agarwal93043844 , enhancing optomechanical cooling Huang79013821 ,
the normal-mode splitting Huang80033807 , controlling the photon blockade
Sarma96053827 ; Shen98023856 ; Shen101013826 , and the increase of atom-cavity
coupling Qin120093601 .

Recently, studying the nonlinear optomechanical interactions in the presence
of a coherent mechanical pump has emerged as an important frontier
Lemonde111053602 ; Liu111083601 ; Mikkelsen96043832 ; Ferretti85033303 . Due
to the existence of nonlinear optomechanical interactions, second-order and
higher-order sidebands are generated in optomechanical systems Xiong86013815 ;
Kronwald111133601 ; Suzuki92033823 ; Jiao18083034 ; Liu717637 ; Fan65850 .
Generation of spectral components at high-order OMIT sidebands is demonstrated
analytically, which may have great potential in precise sensing of charges
Xiong423630 ; Kong95033820 , phonon number Cohen520522 , weak forces
Nunnenkamp111053603 ; Zhao63224211 , single-particle detection Li116 ,
magnetometer Liu712521 , mass sensor Liu99033822 ; Wang106803908 , and high-
order squeezed frequency combs Liu439 . But actually, high-order OMIT
sidebands are generally much weaker than the probe signal, which imposes many
difficulties in detecting and utilizing the second-order sideband. Therefore,
the enhancement and control of second-order sidebands have attracted much
interest. Moreover, by controlling the group delay of the output light field,
which is caused by rapid phase dispersion, slow light or fast light effects
can be achieved Boyd3261074 ; Jiao97013843 ; H5367 ; He351649 ; Li635090 ;
Mirza2725515 ; Liao11698 . The fast and slow light effects of the
optomechanical system have a wide range of applications in optical
communication and interferometry Zimmer92253201 ; Shahriar75053807 . The
hybrid nonlinear optomechanical system provides an important platform for
further study of the tunable slow and fast effect.

For open systems breuer2002 ; Weiss2008 , only if the coupling between the
system and environment is weak, where the characteristic times of the bath are
sufficiently smaller than those of the quantum system under study, the
Markovian approximation is valid. This means that the Markovian approximation
may fail in some cases, e.g., two-state systems, harmonic oscillators, coupled
cavities, etc Chang052105 ; Tan032102 ; Longhi063826 ; Leggett5911987 ;
breuer1032104012009 ; laine810621152010 ; addis900521032014 ;
wibmann860621082012 ; wibmann920421082015 ; shen960338052017 ;
lorenzo880201022013 ; rivas1050504032010 ; luo860441012012 ; wolf1011504022008
; lu820421032010 ; chruscinski1121204042014 ; Zhang063853 ; Zhang19083022 ;
Xiong436053 ; Zhao2729082 ; Triana116183602 ; Cheng430385 ; Cheng623678 ;
Mu46270 ; Li361363 ; Ding111814 ; Sinha124043603 ; Wu103010601 ; Mu94012334 ,
where we need to consider the influences of non-Markovian effects on the
system dynamics. Moreover, we show that the non-Markovian process proves to be
useful in quantum information processing including quantum state engineering,
quantum control, quantum channel capacity caruso8612032014 ; darrigo3502112014
; lofranco900543042014 ; bylicka457202014 ; xue860523042012 , and has been
realized in experiment Xiong2019100 ; Cialdi2019100 ; Tang201297 ;
Groblacher20156 ; Liu20117 ; Hoeppe2012108 ; Xu201082 ; Madsen2011106 ;
Guo2021126 ; Khurana201999 ; Uriri2020101 ; Liu2020102 ; Anderson199347 ;
Li2022129 ; breuer880210022016 ; Vega015001 .

The above two considerations motivate us to explore that how to enhance and
control the second-order OMIT sidebands and group delays in a spinning
resonator with parametric amplifier and non-Markovian effects.

In this paper, we consider the influence of the OPA driven with different
pumping frequencies on the second-order sideband generation in a rotating
optomechanical system, which is coherently driven by a control field and a
probe field. The results show that the second-order sidebands in the rotating
resonator can be greatly enhanced in the presence of the OPA and meanwhile,
remain the nonreciprocal behavior due to the optical Sagnac effect. The
second-order sidebands can be adjusted simultaneously by the pumping frequency
and phase of the field driving the OPA, the gain coefficient of the OPA, the
rotation speed of the resonator, and the incident direction of the input
fields. We compare the differences in efficiency of the second-order sideband
generation when the OPA is driven by different pumping frequencies. Due to the
Sagnac transformation and presence of the OPA, we find that the group delay of
the second-order upper sideband can be tuned by adjusting the nonlinear gain
and phase of the field driving the OPA, the rotation speed of the resonator,
and the incident direction of the input fields in the spinning optomechanical
system. The second-order OMIT sidebands in the spinning resonator are then
generalized to the non-Markovian regimes and compared with the Markovian
approximation in the wideband limit. The influences of the decay from the non-
Markovian environment coupling to an external reservoir on the efficiency of
second-order upper sidebands are also investigated. Our paper indicates the
advantage of using a hybrid nonlinear system, which provides an effective way
to further control and enhance second-order and higher-order sidebands in a
nonreciprocal optical device.

The rest of this paper is organized as follows. In Sec. II, we give the
efficiency of the second-order sideband and its group delay by solving the
Heisenberg-Langevin equations. In Sec. III, we discuss the influence of the
OPA excited by a pump driving with the frequency being the sum of the
frequencies of the strong control field and the weak probe field driving the
resonator on the second-order upper and lower sidebands generation in the
spinning resonator. In Sec. IV, we study the group delay of the second-order
upper sideband. In Sec. V, we show the influence of the OPA on the second-
order sideband generation when the OPA is excited by a pump driving with the
frequency setting to twice the frequency of the strong control field. In Sec.
VI, we extend nonreciprocal second-order sidebands in the spinning resonator
to a non-Markovian bath and compare it with that in the Markovian regime.
Moreover, we also study the influences of the decay from the non-Markovian
environment coupling to an external reservoir on the efficiency of second-
order upper sidebands. Sec. VII is devoted to conclusions.

Figure 1: Schematic diagram of the spinning optomechanical system. A rotating
whispering-gallery-mode (WGM) microresonator (containing an OPA Gerry ;
Clerk821155 ; Nation841 ; Leghtas347853 ; Shen100023814 ; Li100023838  with
the frequency $\omega_{g}$) is coupled to a stationary tapered fiber. The
resonator supports a mechanical mode at frequency ${{\omega_{m}}}$. We fix the
clockwise rotation of the resonator, which leads to that the light circulating
in the resonator experiences a Sagnac-Fizeau shift. (a) ${\Delta_{s}}>0$ and
(b) ${\Delta_{s}}<0$ respond to the control-probe fields come from the left
side and right side, respectively. The nonlinear crystal is pumped by an
additional laser beam to produce parametric amplification. (c) with pump
frequency $\omega_{g}=\omega_{l}+\omega_{p}$ and (d) with pump frequency
$\omega_{g}=2\omega_{l}$ show the level schematic of the optomechanical system
with OPA, where $|{{n_{p}}}\rangle$ and $|{{n_{m}}}\rangle$ denote the number
states of the cavity and the mechanical mode, respectively.

##  II The Model

As schematically shown in Fig. 1(a) and (b), the model we consider is a
rotating whispering-gallery-mode (WGM) microresonator (containing an optical
parametric amplifier), which is coupled to a stationary tapered fiber. The
resonator (driven by a strong control field at frequency ${\omega_{l}}$ and a
weak probe field at frequency ${\omega_{p}}$), with optical resonance
frequency ${{\omega_{0}}}$ and intrinsic loss
${\kappa_{a}}={{{\omega_{0}}}\mathord{\left/{\vphantom{{{\omega_{0}}}Q}}\right.\kern-1.2pt}Q}$
($Q$ is the optical quality factor), supports a mechanical breathing mode
(frequency ${{\omega_{m}}}$ and effective mass $m$). A control laser and a
probe laser are applied to the system via the evanescent coupling of the
optical fiber and resonator, and the field amplitudes are given by
${\varepsilon_{l}}=\sqrt{{P_{l}}/\hbar{\omega_{l}}}$ and
${\varepsilon_{p}}=\sqrt{{P_{p}}/\hbar{\omega_{p}}}$, where ${{P_{l}}}$ and
${{P_{p}}}$ are the control and probe powers, respectively. It is well-known
that due to the rotation, optical mode frequency experiences Sagnac-Fizeau
shift Maayani558569 ; Post39475 ; Malykin431229 , which transforms

$\displaystyle{\omega_{0}}\to{\omega_{0}}+{\Delta_{s}},$ (1)
$\displaystyle{\Delta_{s}}=\frac{{nR\Omega{\omega_{0}}}}{c}\left({1-\frac{1}{{{n^{2}}}}-\frac{\lambda}{n}\frac{{dn}}{{d\lambda}}}\right),$
(2)

where $\Omega=\dot{\phi}$ is the angular velocity of the spinning resonator.
$n$ and $R$ are the refractive index and radius of the resonator,
respectively. $c$ and $\lambda$ are the speed of light and the light
wavelength in a vacuum, respectively. The dispersion term
${{dn}\mathord{\left/{\vphantom{{dn}{d\lambda}}}\right.\kern-1.2pt}{d\lambda}}$
represents a negligibly small relativistic (dispersion) correction in the
Sagnac-Fizeau shift Maayani558569 ; Jiang10064037 . In Eq. (2), the first term
in the parenthesis shows the Sagnac contribution which arises from the
rotation of the resonators, while the two last terms with negative signs take
into account the Fizeau drag due to the light propagation through a moving
resonator medium. As shown in Refs.Agarwal93043844 ; Huang95023844 ; Scully ,
the operating mechanism of the OPA is standard two-photon squeezing. Embedding
the OPA in an optomechanical cavity makes the squeezed state transfer between
a photon of a cavity field and a phonon of mechanical mode, which can amplify
nonlinear optical responses of the system and reduce mechanical thermal noise
and photon shot noise. The Hamiltonian formulation of the system reads

$\hat{H}={{\hat{H}}_{mech}}+{{\hat{H}}_{opt}}+{{\hat{H}}_{OPA}}+{{\hat{H}}_{drive}},$
(3)

with

$\displaystyle{{\hat{H}}_{mech}}=$
$\displaystyle\frac{{{{\hat{p}}^{2}}}}{{2m}}+\frac{1}{2}m\omega_{m}^{2}{{\hat{x}}^{2}}+\frac{{\hat{p}_{\phi}^{2}}}{{2m{{\left({R+\hat{x}}\right)}^{2}}}},$
(4) $\displaystyle{{\hat{H}}_{opt}}=$
$\displaystyle\hbar\left({{\omega_{0}}+{\Delta_{s}}}\right){{\hat{a}}^{\dagger}}\hat{a}-\hbar\xi{{\hat{a}}^{\dagger}}\hat{a}\hat{x},$
$\displaystyle{\hat{H}_{OPA}}=$ $\displaystyle i\hbar
G({\hat{a}^{{\dagger}2}}{e^{i\theta}}{e^{-i{\omega_{g}}t}}-H.c.),$
$\displaystyle{{\hat{H}}_{drive}}=$ $\displaystyle
i\hbar\sqrt{{\kappa_{ex}}}\left({{\varepsilon_{l}}{{\hat{a}}^{\dagger}}{e^{-i{\omega_{l}}t}}+{\varepsilon_{p}}{{\hat{a}}^{\dagger}}{e^{-i{\omega_{p}}t}}-H.c.}\right),$

where ${\hat{p}}$, ${\hat{x}}$, ${\hat{\phi}}$, ${{\hat{p}}_{\phi}}$ describe
the momentum, position, rotation angle, and angular momentum operators, with
commutation relations $[{\hat{x},{\kern 1.0pt}\hat{p}}]=[{\hat{\phi},{\kern
1.0pt}{{\hat{p}}_{\phi}}}]=i\hbar$ Davuluri11264002 . H.c. stands for the
Hermitian conjugate. $\hat{a}\left({{{\hat{a}}^{\dagger}}}\right)$ is the
annihilation (creation) operator of the cavity field with resonance frequency
${{\omega_{0}}}$.
$\xi={{{\omega_{0}}}\mathord{\left/{\vphantom{{{\omega_{0}}}R}}\right.\kern-1.2pt}R}$
is the optomechanical coupling. ${{\hat{H}}_{OPA}}$ describes the coupling of
the intracavity field with the OPA (pump frequency $\omega_{g}$). $G$ is the
nonlinear gain of the OPA, which is proportional to the pump power driving
amplitude. $\theta$ is the phase of the field driving the OPA
Shahidani88053813 . We assume that this OPA with a second-order nonlinearity
crystal is excited by a pump driving with the frequency
$\omega_{g}={{\omega_{l}}+{\omega_{p}}}$ Liu99033822  in Fig. 1(c), so that
the signal light and idler light in OPA have the same frequency
$({\omega_{l}}+{\omega_{p}})/2$ Nation841 ; Leghtas347853 ; Adiyatullin81329 ;
Clerk821155 . ${{\hat{H}}_{drive}}$ describes the interaction of the cavity
field with the control field and that of the cavity field with the probe
field, where ${{\kappa_{ex}}}$ is the loss caused by the resonator-fiber
coupling.

In the rotating frame at the control frequency ${{\omega_{l}}}$, the
Hamiltonian (3) becomes

$\displaystyle{{\hat{H}}_{eff}}=$
$\displaystyle\hbar\left({{\Delta_{0}}-\xi\hat{x}+{\Delta_{s}}}\right){{\hat{a}}^{\dagger}}\hat{a}+\frac{{{{\hat{p}}^{2}}}}{{2m}}+\frac{1}{2}m\omega_{m}^{2}{{\hat{x}}^{2}}$
(5)
$\displaystyle+\frac{{\hat{p}_{\phi}^{2}}}{{2m{{\left({R+\hat{x}}\right)}^{2}}}}+i\hbar
G({\hat{a}^{{\dagger}2}}{e^{-i{\Delta_{p}}t}}{e^{i\theta}}-H.c.)$
$\displaystyle+i\hbar\sqrt{{\kappa_{ex}}}\left[{\left({{\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}}}\right){{\hat{a}}^{\dagger}}-H.c.}\right],$

where ${\Delta_{0}}={\omega_{0}}-{\omega_{l}}$ and
${\Delta_{p}}={\omega_{p}}-{\omega_{l}}{\kern 1.0pt}$. When the control field
is injected at the red-detuned sideband of the cavity resonance
($\Delta_{p}=\omega_{m}$), the transition
$\left|{{n_{p}},{n_{m}}+1}\right\rangle\leftrightarrow\left|{{n_{p}}+1,{n_{m}}}\right\rangle$
occurs. Moreover, $\left|{{n_{p}},{n_{m}}}\right\rangle$ couples with
$\left|{{n_{p}}+1,{n_{m}}}\right\rangle$ through the probe field which is in
resonance with the cavity mode ($\omega_{p}=\omega_{0}$). In this case, the
destructive interference of these two excitation pathways occurs, which leads
to OMIT Weis3301520  in Fig. 1(c) with pump frequency
$\omega_{g}=\omega_{l}+\omega_{p}$ (see Sec. II-IV, Sec. VI) and Fig. 1(d)
with pump frequency $\omega_{g}=2\omega_{l}$ (see Sec. V), where OPA has
almost no influence on the interference paths. With the operator expectation
values defined by $a\equiv\langle{\hat{a}}\rangle$ ,
$x\equiv\langle{\hat{x}}\rangle$ , $\phi\equiv\langle{\hat{\phi}}\rangle$, and
${p_{\phi}}\equiv\langle{{{\hat{p}}_{\phi}}}\rangle$, the Heisenberg-Langevin
equations of the spinning optomechanical system can be derived as

$\displaystyle\dot{a}=-\left[{\kappa+i\left({{\Delta_{0}}-\xi
x+{\Delta_{s}}}\right)}\right]a$
$\displaystyle\qquad+\sqrt{{\kappa_{ex}}}\left({{\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}}}\right)+2G{a^{*}}{e^{i\theta}}{e^{-i{\Delta_{p}}t}},$
(6) $\displaystyle
m(\ddot{x}+{\Gamma_{m}}\dot{x}+\omega_{m}^{2}x)=\hbar\xi{a^{*}}a+\frac{{p_{\phi}^{2}}}{{m{R^{3}}}},$
(7) $\displaystyle\dot{\phi}=\frac{{{p_{\phi}}}}{{m{R^{2}}}},$ (8)
$\displaystyle{{\dot{p}}_{\phi}}=0,$ (9)

where $\kappa=({\kappa_{a}}+{\kappa_{ex}})/2$ and $\Gamma_{m}$ are the
dissipations of the cavity and the damping of the mechanical mode,
respectively. The derivation of Eqs. (6)-(9) can be found in Appendix.
Focusing on the mean response of the system to the probe field, we write the
operators for their expectation values by means of the mean-field
approximation and safely ignore the quantum noise terms with strong driving
conditions.

In this case, we assume the control field is much stronger than the probe
field (${\varepsilon_{l}}\gg{\varepsilon_{p}}$), which induces that we can use
the perturbation method to deal with Eqs. (6)-(9). The control field provides
a steady-state solution of the system, while the probe field is treated as the
perturbation of the steady state. We then follow the standard procedure, which
decomposes the expectation value of all operators as a sum of their steady-
state value and small fluctuations around the steady-state value Weis3301520 ;
Xiong86013815

$\displaystyle a=$
$\displaystyle{a_{s}}+A_{1}^{+}{e^{-i{\Delta_{p}}t}}+A_{1}^{-}{e^{i{\Delta_{p}}t}}+A_{2}^{+}{e^{-2i{\Delta_{p}}t}}+A_{2}^{-}{e^{2i{\Delta_{p}}t}},$
(10) $\displaystyle x=$
$\displaystyle{x_{s}}+X_{1}^{+}{e^{-i{\Delta_{p}}t}}+X_{1}^{-}{e^{i{\Delta_{p}}t}}+X_{2}^{+}{e^{-2i{\Delta_{p}}t}}+X_{2}^{-}{e^{2i{\Delta_{p}}t}},$

in which $A_{2}^{+}$ ($A_{2}^{-}$) is the amplitude of second-order upper
(lower) sideband and corresponds to the responses at the original frequencies
$2{\omega_{p}}-{\omega_{l}}$ ($3{\omega_{l}}-2{\omega_{p}}$). We are committed
to the fundamental OMIT and its second-order sideband process so that the
higher-order sidebands in Eq. (10) are ignored. By substituting Eq. (10) into
Eqs. (6)-(9) and comparing the coefficients of the same order, the steady-
state solutions are obtained as

$\displaystyle{a_{s}}=$
$\displaystyle\frac{{\sqrt{{\kappa_{ex}}}{\varepsilon_{l}}}}{{\kappa+i\Delta}},$
(11) $\displaystyle{x_{s}}=$
$\displaystyle\frac{{\hbar\xi{{\left|{{a_{s}}}\right|}^{2}}}}{{m\omega_{m}^{2}}}+R{\left({\frac{\Omega}{{{\omega_{m}}}}}\right)^{2}},$

where $\Delta{\rm{=}}{\Delta_{0}}-\xi{x_{s}}+{\Delta_{s}}$, and
$\Omega{\rm{=}}{{d\phi}\mathord{\left/{\vphantom{{d\theta}{dt}}}\right.\kern-1.2pt}{dt}}$
is the angular velocity of the spinning resonator. It is clear that the
revolving speed of the resonator and Sagnac-Fizeau shift ${\Delta_{s}}$ affect
the values of both the mechanical displacement ${x_{s}}$ and intracavity
photon number ${\left|{{a_{s}}}\right|^{2}}$. Substituting Eq. (10) into Eqs.
(6)-(9), we gain six algebra equations, which can be divided into two groups.
The first group describes the linear response of the probe field

$\begin{split}{\sigma_{1}}\left({{\Delta_{p}}}\right)A_{1}^{+}&=i\xi{a_{s}}X_{1}^{+}+2G{e^{i\theta}}a_{s}^{*}+\sqrt{{\kappa_{ex}}}{\varepsilon_{p}},\\\
{\sigma_{2}}\left({{\Delta_{p}}}\right)A_{1}^{-*}&=-i\xi
a_{s}^{*}X_{1}^{+},\\\
\chi\left({{\Delta_{p}}}\right)X_{1}^{+}&=\hbar\xi({a_{s}}A_{1}^{-*}+a_{s}^{*}A_{1}^{+}),\end{split}$
(12)

while the second group corresponds to the second-order sideband process

$\begin{split}{\sigma_{1}}(2{\Delta_{p}})A_{2}^{+}&=i\xi({a_{s}}X_{2}^{+}+A_{1}^{+}X_{1}^{+})+2G{e^{i\theta}}A_{1}^{-*},\\\
{\sigma_{2}}\left({2{\Delta_{p}}}\right)A_{2}^{-*}&=-i\xi(a_{s}^{*}X_{2}^{+}+A_{1}^{-*}X_{1}^{+}),\\\
\chi\left({2{\Delta_{p}}}\right)X_{2}^{+}&=\hbar\xi(a_{s}^{*}A_{2}^{+}+{a_{s}}A_{2}^{-*}+A_{1}^{-*}A_{1}^{+}),\end{split}$
(13)

with

$\displaystyle{\sigma_{1}}\left({n{\Delta_{p}}}\right)$ $\displaystyle=$
$\displaystyle\kappa+i\Delta-in{\Delta_{p}},$
$\displaystyle{\sigma_{2}}\left({n{\Delta_{p}}}\right)$ $\displaystyle=$
$\displaystyle\kappa-i\Delta-in{\Delta_{p}},$
$\displaystyle\chi\left({n{\Delta_{p}}}\right)$ $\displaystyle=$
$\displaystyle m(\omega_{m}^{2}-i{\Gamma_{m}}n{\Delta_{p}}-\Delta_{p}^{2}).$

Moreover, we can easily get the linear and second-order nonlinear responses of
the system

$\begin{split}A_{1}^{+}&=\frac{{D+{\sigma_{2}}\left({{\Delta_{p}}}\right)\chi\left({{\Delta_{p}}}\right)}}{{{f_{3}}\left({{\Delta_{p}}}\right)}}(\sqrt{{\kappa_{ex}}}{\varepsilon_{p}}+2G{e^{i\theta}}a_{s}^{*}),\\\
X_{1}^{+}&=\frac{{\hbar\xi
a_{s}^{*}{\sigma_{2}}\left({{\Delta_{p}}}\right)}}{{D+{\sigma_{2}}\left({{\Delta_{p}}}\right)\chi\left({{\Delta_{p}}}\right)}}A_{1}^{+},\\\
A_{1}^{-*}&=\frac{{-i\xi
a_{s}^{*}}}{{{\sigma_{2}}\left({{\Delta_{p}}}\right)}}X_{1}^{+},\end{split}$
(14)

and

$\begin{split}A_{2}^{+}&=\frac{{-D{\xi^{2}}{a_{s}}X_{1}^{+2}+i\xi{f_{1}}A_{1}^{+}X_{1}^{+}-2i\xi
G{e^{i\theta}}a_{s}^{*}{f_{2}}X_{1}^{+}}}{{{\sigma_{2}}\left({{\Delta_{p}}}\right){f_{3}}\left({2{\Delta_{p}}}\right)}},\\\
X_{2}^{+}&=\frac{{\hbar\xi[{\sigma_{2}}(2{\Delta_{p}})a_{s}^{*}A_{2}^{+}+{\sigma_{2}}(2{\Delta_{p}})A_{1}^{+}A_{1}^{-*}-i\xi{a_{s}}A_{1}^{-*}X_{1}^{+}]}}{{{f_{2}}}},\\\
A_{2}^{-}&=\frac{{i\xi}}{{{\sigma_{2}}{{\left({2{\Delta_{p}}}\right)}^{*}}}}({a_{s}}X_{2}^{-}+A_{1}^{-}X_{1}^{-}),\end{split}$
(15)

where

$\begin{split}D&=i\hbar{\xi^{2}}{\left|{{a_{s}}}\right|^{2}},\\\
{f_{1}}&=iD{\Delta_{p}}+{\sigma_{2}}\left({{\Delta_{p}}}\right){\sigma_{2}}\left({2{\Delta_{p}}}\right)\chi\left({2{\Delta_{p}}}\right),\\\
{f_{2}}&=D+{\sigma_{2}}\left({2{\Delta_{p}}}\right)\chi\left({2{\Delta_{p}}}\right),\\\
{f_{3}}\left({n{\Delta_{p}}}\right)&=2iD\Delta+{\sigma_{1}}(n{\Delta_{p}}){\sigma_{2}}\left({n{\Delta_{p}}}\right)\chi\left({n{\Delta_{p}}}\right).\end{split}$

By using the standard input-output relations, i.e.,

${a_{out}}(t)={a_{in}}(t)-\sqrt{{\kappa_{ex}}}a(t),$ (16)

we obtain the expectation value of the output field of this system

$\displaystyle{a_{out}}(t)=$
$\displaystyle{C_{1}}{e^{-i{\omega_{l}}t}}+{C_{2}}{e^{-i{\omega_{p}}t}}-\sqrt{{\kappa_{ex}}}A_{1}^{-}{e^{-i(2{\omega_{l}}-{\omega_{p}})t}}$
(17)
$\displaystyle-\sqrt{{\kappa_{ex}}}A_{2}^{+}{e^{-i(2{\omega_{p}}-{\omega_{l}})t}}-\sqrt{{\kappa_{ex}}}A_{2}^{-}{e^{-i(3{\omega_{l}}-2{\omega_{p}})t}},$

where ${C_{1}}={\varepsilon_{l}}-\sqrt{{\kappa_{ex}}}a{}_{s}$ and
${C_{2}}={\varepsilon_{p}}-\sqrt{{\kappa_{ex}}}A_{1}^{+}$. The first term of
Eq. (17) denotes the output with control frequency ${{\omega_{l}}}$, while the
second and third terms describe the anti-Stokes and Stokes fields,
respectively. The terms
$-\sqrt{{\kappa_{ex}}}A_{2}^{+}{e^{-i(2{\omega_{p}}-{\omega_{l}})t}}$ and
$-\sqrt{{\kappa_{ex}}}A_{2}^{-}{e^{-i(3{\omega_{l}}-2{\omega_{p}})t}}$ are
concerned in the second-order upper and lower sidebands Xiong86013815 .

Subsequently, we introduce the dimensionless quantity to define the efficiency
of the second-order upper and lower sidebands Xiong86013815 ; Teufel471204

$\displaystyle{\eta_{1}}=\left|{-\frac{{\sqrt{{\kappa_{ex}}}A_{2}^{+}}}{{{\varepsilon_{p}}}}}\right|,$
(18)
$\displaystyle{\eta_{2}}=\left|{-\frac{{\sqrt{{\kappa_{ex}}}A_{2}^{-}}}{{{\varepsilon_{p}}}}}\right|,$
(19)

where the amplitude of the probe pulse is treated as a basic scale to gauge
the amplitudes of the output sidebands ${\eta_{1}}$ and ${\eta_{2}}$. The
associated group delay of the second-order upper sideband turns out to be
Safavi47269 ; Lu1 ; Lu2

${\tau_{1}}={\left.{\frac{{d{\kern
1.0pt}\arg\left({-\frac{{\sqrt{{\kappa_{ex}}}A_{2}^{+}}}{{{\varepsilon_{p}}}}}\right)}}{{2d{\Delta_{p}}}}}\right|_{{\Delta_{p}}={\omega_{m}}}}.$
(20)

A positive group delay (${\tau_{1}}>0$) corresponds to slow light phenomenon,
while a negative group delay (${\tau_{1}}<0$) corresponds to fast light
phenomenon Safavi47269 ; Milonni .

Figure 2: The efficiency ${\eta_{1}}$ of the second-order upper sideband
generation as a function of ${\Delta_{p}}$ for different values of $\Omega$
and incident directions of light, where the nonlinear gain and phase of the
probe field of the OPA are fixed as (a) $G=0,\theta=0$; (b)
$G=0.2\kappa,\theta=0$; (c) $G=0.2\kappa,\theta=3\pi/2$. ${\eta_{1}}$ varies
with ${\Delta_{p}}$ and $\Omega$ under different values (d) $G=0,\theta=0$;
(e) $G=0.2\kappa,\theta=0$; (f) $G=0.2\kappa,\theta=3\pi/2$. Other parameters
are ${P_{p}}=0.05{P_{l}}$, ${P_{l}}=1$ mW, $\lambda=1550$ nm, $R=0.25$ mm,
$m=25$ ng, $n=1.44$, $Q={\omega_{0}}/\kappa=4.5\times{10^{7}}$,
${\omega_{m}}=100$ MHz, ${\Gamma_{m}}=0.1$ MHz,
${\kappa_{a}}={\kappa_{ex}}={\omega_{0}}/Q$, ${P_{p}}=0.05{P_{l}}$, and
${\Delta_{0}}={\omega_{m}}$, respectively. With the parameters, we obtain the
Sagnac-Fizeau shift $\Delta_{s}$=($15.082$MHz, $0$, $15.082$MHz) or
$\Delta_{s}/\omega_{m}$=($0.1508$, $0$, $-0.1508$), which corresponds to the
angular velocity $\Omega=$ ($20$kHz, 0, $-20$kHz) of the cavity.

##  III Results and discussions

In our numerical simulations, to demonstrate that the observation of the
second-order sidebands in a resonator assisted by OPA is within current
experimental reach, we calculate Eqs. (18)-(20) with parameters from
Refs.Maayani558569 ; Righini34435 ; Zhang1321 : $\lambda=1550$ nm, $R=0.25$ mm
(the resonator radius), $m=25$ ng, $n=1.44$,
$Q={\omega_{0}}/\kappa=4.5\times{10^{7}}$, ${\omega_{m}}=100$ MHz,
${\Gamma_{m}}=0.1$ MHz, ${\kappa_{a}}={\kappa_{ex}}={\omega_{0}}/Q$,
${P_{p}}=0.05{P_{l}}$, and ${\Delta_{0}}={\omega_{m}}$, respectively. We
rotate the resonator clockwise, where $\Omega>0$ stands for the light coming
from the left-hand side and $\Omega<0$ denotes the light coming from the
right-hand side.

To see the influence of resonator rotation and OPA on the second-order
sideband generation, the efficiency of second-order upper sideband generation
is investigated as a function of frequency ${\Delta_{p}}/{\omega_{m}}$ shown
in Fig. 2. In Fig. 2(a), we discuss that the efficiency ${\eta_{1}}$ of the
second-order upper sideband varies with ${\Delta_{p}}$ without the
participation of OPA, i.e., the nonlinear gain of the OPA $G=0$, the phase of
the field driving the OPA $\theta=0$. Under the resonator stationary, we find
two located peaks of second-order sideband spectra and a local minimum near
the resonance condition ${\Delta_{p}}={\omega_{m}}$. By spinning the
resonator, the peak position of ${\eta_{1}}$ has different moves when the
driving fields come from different directions. By adjusting the frequency
${\Delta_{p}}/{\omega_{m}}$, we can get enhanced efficiency of the second-
order sideband while driving the resonator from one direction and suppressed
efficiency while driving from the opposite direction. For example, within
${\Delta_{p}}/{\omega_{m}}$ in the range from $0.99$ to $1$, ${\eta_{1}}$ is
enhanced in the case of $\Omega>0$, while it is suppressed in the case of
$\Omega<0$. Obviously, this spinning-induced direction-dependent nonreciprocal
behavior can be attributed to the optical Sagnac effect induced by a spinning
resonator. As shown in Fig. 2(b), the efficiency ${\eta_{1}}$ gets larger in
the presence of OPA. To be more specific, for $G=0.2\kappa,\theta=0$ and
$\Omega=20$ kHz, the efficiency ${\eta_{1}}$ can increase from
${\rm{19}}{\rm{.5\%}}$ to $27.2{\rm{\%}}$ at ${\Delta_{p}}=0.997{\omega_{m}}$.
When the system is driven from the right, i.e., $\Omega=-20$ kHz, the
efficiency ${\eta_{1}}$ also can increase from $12.4{\rm{\%}}$ to
$21.6{\rm{\%}}$ at ${\Delta_{p}}=1.004{\omega_{m}}$. Fig. 2(c) shows that the
efficiency ${\eta_{1}}$ can also be adjusted by tuning $\theta$. What can be
seen clearly is when $\theta$ changes from $\theta=0$ to $\theta=3\pi/2$, in
the case of $G=0.2\kappa$ and $\Omega=20$ kHz, the maximum value of
${\eta_{1}}$ increases to $37.4{\rm{\%}}$. In the case of $\Omega=-20$ kHz,
the maximum value increases to $29.2{\rm{\%}}$. We see that the efficiency of
the second-order upper sideband is sensitive to the variation of the nonlinear
gain of the OPA and phase of the field driving the OPA, which indicates the
advantage of using a hybrid nonlinear system. According to Eqs. (14)-(15),
such phenomena coming from the amplitudes of second-order sidebands
$A_{2}^{+}$ and $A_{2}^{-}$ are related directly to the Sagnac-Fizeau shift
and OPA. With the purpose of seeing the influence of the OPA on the second-
order sideband generation more clearly, the efficiency ${\eta_{1}}$ as a
function of both ${\Delta_{p}}$ and $\Omega$ is shown in Fig. 2(d)-(f).

Figure 3: The efficiency ${\eta_{1}}$ of the second-order upper sideband
generation as a function of the probe-pulsed detuning ${\Delta_{p}}$ for
different (a) nonlinear gain $G$ of the OPA with $\theta=0$ and (c) phase
$\theta$ with $G=0.2\kappa$. (b) ${\eta_{1}}$ varies with ${\Delta_{p}}$ and
$G$ under $\theta=0$. (d) ${\eta_{1}}$ varies with ${\Delta_{p}}$ and $\theta$
under $G=0.2\kappa$. The angular velocity of the spinning resonator is fixed
at $\Omega=-20$ kHz. Other parameters are the same as Fig. 2.

To explore the role of OPA in this resonator, we illustrate the efficiency
${\eta_{1}}$ of the second-order upper sideband versus the probe-pulsed
detuning ${\Delta_{p}}$ with different nonlinear gain $G$ of the OPA and phase
$\theta$ of the field driving the OPA, when the system is driven from the
right-hand side ($\Omega=-20$ kHz) in Fig. 3. We find when the nonlinear gain
$G$ of the OPA increases from $0$ to $G=0.6\kappa$, the efficiency
${\eta_{1}}$ can be significantly enhanced in Fig. 3(a). The enhancement
effect at the probe-pulsed detuning ${\Delta_{p}}/{\omega_{m}}<1$ is much
weaker than at ${\Delta_{p}}/{\omega_{m}}>1$. Fig. 3(c) shows that the second-
order sideband behavior of the output field can also be adjusted by tuning
$\theta$. In the case of $G=0.2\kappa$, we find that compared with $\theta=0$,
both $\theta=\pi/2$ and $\theta=\pi$ result in lower efficiency ${\eta_{1}}$
of the second-order upper sideband, but $\theta=3\pi/2$ leads to enhanced
efficiency. In Fig. 3(d), ${\eta_{1}}$ as a function of detuning
${\Delta_{p}}$ and the phase $\theta$ of the OPA is plotted. In the range
shown, the maximum value of efficiency ${\eta_{1}}$ is about $30.2\%$ at
$\theta=1.6\pi$ and ${\Delta_{p}}=1.004{\omega_{m}}$. Specifically, the
efficiency is enhanced when $\theta\in(1.6\pi,{\kern 1.0pt}{\kern 1.0pt}2\pi)$
and suppressed at other values. Besides, as is illustrated in Fig. 3(b) and
(d), regardless of what nonlinear gain $G$ and $\theta$ is to increase, the
located maximums of the efficiency ${\eta_{1}}$ are still located at the same
position of the probe-pulsed detuning. This phenomenon can be explained by
Refs.Weis3301520 ; Xiong86013815 , which shows there are some connections
between OMIT and the second-order sideband process. When OMIT occurs, the
second-order sideband process is subdued. The linewidth of the OMIT window is
related to the intracavity photon number

${\Gamma_{OMIT}}\approx{\Gamma_{\rm{m}}}+\frac{{{\xi^{2}}x_{zpf}^{2}}}{\kappa}{\left|{{a_{s}}}\right|^{2}},$
(21)

where ${x_{zpf}}=\sqrt{\hbar/2m{\omega_{\rm{m}}}}$. By perturbation theory, we
can get the intracavity photon number ${\left|{{a_{s}}}\right|^{2}}$ in Eq.
(11), which is independent of other perturbation terms such as probe pulse and
nonlinear gain of the OPA. That is to say, the positions of these local
maximums of sideband spectra only depend on the intrinsic structural
parameters of an optomechanical system and intensity of the control field. As
a result, the OPA not only improves the sideband efficiency of the second-
order sideband but also keeps the locality of maximum values of the sideband
efficiency.

Figure 4: The efficiency ${\eta_{2}}$ of the second-order lower sideband
generation as a function of ${\Delta_{p}}$ for different values of $\Omega$
and incident directions of light, where the nonlinear gain and phase of the
probe field of the OPA are fixed as (a) $G=0,\theta=0$; (b)
$G=0.2\kappa,\theta=0$; (c) $G=0.2\kappa,\theta=3\pi/2$. ${\eta_{2}}$ varies
with ${\Delta_{p}}$ and $\Omega$ under different values (d) $G=0,\theta=0$;
(e) $G=0.2\kappa,\theta=0$; (f) $G=0.2\kappa,\theta=3\pi/2$. Other parameters
are the same as Fig. 2.

In Fig. 4, we discuss the influence of resonator rotation and OPA on the
second-order lower sideband generation. As shown in Fig. 4(a), unlike the
second-order upper sideband, the second-order lower sideband has no local
minimum but only one peak. The efficiency is much smaller than the second-
order upper sideband. In detail, with neither resonator rotation nor the OPA
drive, ($G=0$, $\Omega=0$), both peaks of ${\eta_{1}}$ are about $19.6\%$ and
the peak of ${\eta_{2}}$ is only $0.82\%$. Furthermore, the second-order lower
sideband exhibits non-reciprocal characteristics due to the rotation of the
resonator, which is more pronounced at ${\Delta_{p}}/{\omega_{m}}>1$. In
detail, compared with the stationary resonator (i.e., no spinning with
$\Omega=0$), the spinning resonator increases for $\Omega=-20$ kHz, while it
decreases for $\Omega=20$ kHz at ${\Delta_{p}}/{\omega_{m}}>1$ in Fig. 4(a).
In Fig. 4(d), we find that for the same resonator speed, the enhancement
effect is more pronounced when the device is driven from the right side
($\Omega<0$) than from the left side ($\Omega>0$). For example, for
$\Omega=-60$ kHz, the maximum value of ${\eta_{2}}$ is $3.04\%$ at
${\Delta_{p}}/{\omega_{m}}=1.003$. For $\Omega=60$ kHz, the maximum value of
${\eta_{2}}$ is $0.97\%$ at ${\Delta_{p}}/{\omega_{m}}=0.999$. In Fig. 4(b)
and (c), as with the second-order upper sideband, the presence of OPA
significantly improves the efficiency of the second-order lower sideband,
which also keeps the locality of maximum values of the sideband efficiency. In
detail, for $\Omega=-20$ kHz, when the nonlinear gain $G$ of OPA increases
from 0 to $0.2\kappa$, the maximum value of ${\eta_{2}}$ increases from
$1.08\%$ to $1.75\%$ at ${\Delta_{p}}/{\omega_{m}}=1.001$. Besides, when the
phase $\theta$ of the OPA increases from 0 to $3\pi/2$, the maximum value of
${\eta_{2}}$ can be increased to $2.51\%$, which is more than twice the value
without OPA.

We show that the presence of OPA only causes a change in the peak of
${\eta_{1}}$ and has almost no influence on asymmetry (see black-solid line in
Fig. 2(a)-(c) for $\Omega=0$, and black-solid line in Fig. 4(a)-(c) for
$\Omega=0$).

Without the OPA ($G=0$), the asymmetric line shape of ${\eta_{1}}$ with regard
to ${\Delta_{p}}={\omega_{m}}$ and the ${\eta_{2}}$ peak being not exactly at
${\Delta_{p}}={\omega_{m}}$ come from the spinning of the resonator. In this
case, the mean mechanical displacement ${x_{s}}$ in Eq. (11) is made up of two
terms: the first term is proportional to the intracavity photon number
$|{a_{s}}|^{2}$, which is closely related to the Sagnac-Fizeau shift
${\Delta_{s}}=\frac{{nR\Omega{\omega_{0}}}}{c}({1-\frac{1}{{{n^{2}}}}-\frac{\lambda}{n}\frac{{dn}}{{d\lambda}}})$
in Eq. (2) or, equivalently, very sensitive to the angular velocity $\Omega$
of resonator and incident direction of input fields, thus giving rise to the
nonreciprocal behavior. The second term $R{(\Omega/{\omega_{m}})^{2}}$ of
${x_{s}}$ makes the mechanical displacement larger due to the rotation. The
existence of these two terms together affects the second-order upper and lower
sidebands in Eqs. (18) and (19), which lead to the asymmetry of ${\eta_{1}}$
with regard to ${\Delta_{p}}={\omega_{m}}$ and the ${\eta_{2}}$ peak being not
exactly at ${\Delta_{p}}={\omega_{m}}$ shown in Fig. 2 to Fig. 4.

In this case, $R{(\Omega/{\omega_{m}})^{2}}$ of ${x_{s}}$ in Eq. (11)
originates from an extra term in the Hamiltonian of our model due to the
rotation, i.e., the rotational kinetic energy term
$\hat{p}_{\phi}^{2}/[2m{({R+\hat{x}})^{2}}]$ in Eq. (4), which is different
from the usual situation in Ref.Xiong86013815 . Since $x/R\ll 1$
($x=\langle{\hat{x}}\rangle$ denotes the expectation value of $\hat{x}$), the
term $\hat{p}_{\phi}^{2}/[2m{({R+\hat{x}})^{2}}]$ is approximately equal to
$-\hat{p}_{\phi}^{2}\hat{x}/(m{R^{3}})+\hat{p}_{\phi}^{2}/(2m{R^{2}})$
(neglecting second and higher order small quantities about $x/R$). This means
that there is an extra force $-\hat{p}_{\phi}^{2}\hat{x}/(m{R^{3}})$ exerted
on the mechanical mode making it deviate from its original equilibrium
position.

Figure 5: The efficiency ${\eta_{1}}$ of the second-order upper sideband
generation as a function of the nonlinear gain $G$ of OPA for different probe-
pulsed detuning ${\Delta_{p}}$, where $\theta=0$ and $\Omega=0$. Other
parameters are the same as Fig. 2.

To clearly see the influence of the nonlinear gain $G$ of OPA on the second-
order sideband generation, the efficiency ${\eta_{1}}$ is investigated as a
function of the nonlinear gain $G$ for different probe-pulsed detuning
${\Delta_{p}}$, as shown in Fig. 5. In detail, when $G$ increases from $0$ to
$0.6\kappa$ in the case of ${\Delta_{p}}/{\omega_{m}}=1.002$, the system
provides an enhancement of more than five times for the sideband efficiency
${\eta_{1}}$. In general, with the nonlinear gain $G$ increasing, the
efficiency ${\eta_{1}}$ of the second-order upper sideband generation
increases obviously. The reason is that when the OPA is pumped at
$\omega_{g}={\omega_{l}}+{\omega_{p}}$, i.e., twice the frequency of the anti-
Stokes field, the parametric frequency conversion between this anti-Stokes
field and phonon mode can provide another way to generate an optical second-
order sideband, leading to the enhancement of a second-order sideband.

##  IV Tunable slow and fast light

Figure 6: Optical group delay of the second-order upper sideband ${\tau_{1}}$
is plotted as a function of ${P_{l}}$ with different values of $\Omega$ and
incident directions of light (a) without OPA and (b) in the presence of OPA
effect at $G=0.4\kappa$ and $\theta=0$. ${\tau_{1}}$ is plotted as a function
of ${P_{l}}$ with different (c) nonlinear gain $G$ and (d) the phase $\theta$
of the field driving the OPA, where $\Omega=0$. Other parameters are the same
as Fig. 2.

We know the slow light effect is an important result of OMIT, which can be
described by the optical group delay Safavi47269 ; He351649 ; Li635090 ;
Mirza2725515 ; Liao11698 . It is similar to that of electromagnetically
induced transparency, in the region of the narrow transparency window the
rapid phase dispersion can cause the group delay given by Eq. (20). A positive
group delay (${\tau_{1}}>0$) corresponds to slow light propagation and a
negative group delay (${\tau_{1}}<0$) denotes fast light propagation.

In the previous work Safavi47269 ; Milonni , it has been demonstrated that the
delay of the transmitted light is only relevant to the pump power in a
conventional optomechanical system. In our model, we see clearly from Fig. 6
that the delay time of the second-order upper sideband can be adjusted not
only by tuning the speed and direction of rotation of the resonator but also
by adjusting the nonlinear gain of the OPA and phase of the field driving the
OPA. In Fig. 6(a) and (b), we investigate the group delay of the second-order
upper sideband ${\tau_{1}}$ as a function of control laser power ${P_{l}}$ for
different $\Omega$. We find that when the resonator is stationary
($\Omega=0$), with the power increasing, ${\tau_{1}}$ tends to advance and
even switches into fast light. However, in the presence of resonator rotation,
the delay time of the second-order upper sideband will be prolonged at high
control powers, which is useful for storage. In detail, as shown in Fig. 6(a),
for a resonator speed of $20$ kHz, the group delay time ${\tau_{1}}$ increases
when the resonator is driven from the right side ($\Omega=-20$ kHz) and
decreases when the resonator is driven from the left side ($\Omega=20$ kHz).
The group delay can still reach the conversion from fast light to slow light
at this point. Increasing the resonator speed to $40$ kHz, at high control
power, when the resonator is driven from the right side ($\Omega=-40$ kHz),
the group delay of the second-order sideband is always positive, i.e., slow
light is obtained. When the resonator is driven from the left side
($\Omega=40$ kHz), the group delay is always negative and fast light can be
obtained. At this point, the switching between fast and slow light disappears.
In Fig. 6(b), we show the results of group delay ${\tau_{1}}$ versus control
laser power $P_{l}$ in the presence of OPA. In the low power range, the
addition of OPA increases the value of ${\tau_{1}}$. More interestingly, at
$\Omega=-40$ kHz, the fast and slow light conversion behavior of the group
delay disappears, where only slow light effect is obtained.

Now we discuss the influence of the presence of OPA on the delay time of the
second-order sideband. In Fig. 6(c) and (d), we display the group delay
${\tau_{1}}$ as a function of the control power ${P_{l}}$ for different
parameters of nonlinear gain $G$ and phase $\theta$ of the field driving the
OPA, where the resonator is stationary. When the OPA is considered in the
optomechanical system, as is expected, the delay time of the second-order
upper sideband generation obviously increases with the increasing power. With
the nonlinear gain $G$ increasing from $0$ to $0.4\kappa$, the group delay
${\tau_{1}}$ accordingly increases, while the trend of switching between fast
and slow light effects remains unchanged. In Fig. 6(d), we see that the
${\tau_{1}}$ is sensitive to the variation of the phase of the OPA. When
$\theta=\pi/2$, ${\tau_{1}}$ exhibits a significant transition from fast to
slow light, in other words the delay time significantly reduces at low power
and increases at high power. Interestingly, for $\theta=3\pi/2$, the valley of
the ${\tau_{1}}$ disappears in the low power range, where the group delay
exhibits a fast light effect (${\tau_{1}}<0$) in the high power range.
Physically, from Eq. (15), when the OPA is added inside the optomechanical
coupled system, the quantum interference effect between the probe field and
second-order sideband process is related directly to the phase of the OPA, so
that the optical-response properties for the probe field become phase-
sensitive.

Figure 7: The group delay of the second-order upper sideband ${\tau_{1}}$
varies with the spinning angular velocity $\left|\Omega\right|$ at $\Omega>0$
and $\Omega<0$, where $G=0.4\kappa$ and $\theta=0$. The power of the control
field ${P_{l}}$ is $1$ mW. Other parameters are the same as Fig. 6.

As shown in Fig. 7, the group delay ${\tau_{1}}$ varies with the rotation
speed of the resonator $\left|\Omega\right|$ at a fixed control power, where
the red sideband ${{\Delta_{p}}={\omega_{m}}}$ is also presented. We find the
group delay can achieve the transition from fast to slow light regardless of
the direction of incidence of the input fields but with very significant
differences. If $\Omega>0$ (the driving fields come from the left-hand side of
the fiber), when the rotation speed reaches $101$ kHz, the group delay
${\tau_{1}}$ experiences the conversion from ${\tau_{1}}<0$ to ${\tau_{1}}>0$.
However, if $\Omega<0$ (driving from the right-hand side of the fiber), when
the rotation speed reaches $30$ kHz, ${\tau_{1}}$ experiences the conversion
from ${\tau_{1}}<0$ to ${\tau_{1}}>0$. Therefore, we realize the conversion
between the fast light and slow light by controlling the incident direction of
the input fields in the spinning system.

Figure 8: (a) The group delay of the second-order upper sideband ${\tau_{1}}$
varies with ${P_{l}}$ and $\Omega$, where $G=0$ and $\theta=0$. (b)
${\tau_{1}}$ varies with ${P_{l}}$ and $\theta$ at $G=0.4\kappa$ and
$\Omega=0$. The black curves correspond to ${\tau_{1}}=0$. Other parameters
are the same as Fig. 6.

In the above discussion, we see that the group delay of the second-order upper
sideband is sensitive to the variation of the rotation speed of the resonator,
the direction of incidence of the input fields, and the phase of the field
driving the OPA. In Fig. 8(a), the group delay ${\tau_{1}}$ of the second-
order upper sideband is plotted as the function of control power ${P_{l}}$ and
the rotation speed of the resonator $\Omega$. In Fig. 8(b), ${\tau_{1}}$ is
plotted as the function of control power ${P_{l}}$ and the phase $\theta$ of
the field driving the OPA. The black curves correspond to ${\tau_{1}}=0$. In
this case, we can obtain the slow light effect or fast light effect by
properly selecting the values of ${P_{l}}$, $\Omega$, and $\theta$. Moreover,
a tunable switch from fast to slow light can be realized by adjusting their
values.

##  V Influence of changing the driving frequency of OPA on the efficiency

The optical degenerate parametric amplifier (OPA), a second-order optical
crystal in nature, can generate pairs of down-converted photons and show
nearly perfect single or dual squeezing Gerry ; Li100023838 ; Clerk821155 ;
Nation841 ; Leghtas347853 ; Shen100023814 . As we all know, placing an OPA
pumped by an external laser in the optomechanical cavity can modulate the
optomechanical coupling, which can lead to optical amplification directly
Adamyan92053818 . We can discuss the influence of different pump frequencies
of the driving OPA on the sidebands and compare the amplification of the
second-order sidebands in both cases. Now, we vary the frequency of the laser
field driving the OPA, so that the OPA is excited by a pump drive with the
frequency $\omega_{g}=2{\omega_{l}}$ Li100023838  in Fig. 1(d). The pump
photon with frequency $\omega_{g}=2{\omega_{l}}$ is down-converted into an
identical pair of photons with frequency ${\omega_{l}}$ after passing through
the second-order nonlinearity crystal. ${{\hat{H}}_{OPA}}$ reads

${\hat{H}_{OPA}}=i\hbar
G({\hat{a}^{{\dagger}2}}{e^{i\theta}}{e^{-2i{\omega_{l}}t}}-H.c.).$ (22)

The total Hamiltonian of the system in the rotating frame at the laser
frequency ${{\omega_{l}}}$ is given by

$\displaystyle{{\hat{H}}_{eff}}=$
$\displaystyle\hbar\left({{\Delta_{0}}-\xi\hat{x}+{\Delta_{s}}}\right){{\hat{a}}^{\dagger}}\hat{a}+\frac{{{{\hat{p}}^{2}}}}{{2m}}+\frac{1}{2}m\omega_{m}^{2}{{\hat{x}}^{2}}$
(23)
$\displaystyle+\frac{{\hat{p}_{\phi}^{2}}}{{2m{{\left({R+\hat{x}}\right)}^{2}}}}+i\hbar
G({\hat{a}^{{\dagger}2}}{e^{i\theta}}-H.c.)$
$\displaystyle+i\hbar\sqrt{{\kappa_{ex}}}\left[{\left({{\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}}}\right){{\hat{a}}^{\dagger}}-H.c.}\right].$

We can get the equations of motion

$\displaystyle\dot{a}=-\left[{\kappa+i\left({{\Delta_{0}}-\xi
x+{\Delta_{s}}}\right)}\right]a$
$\displaystyle\qquad+\sqrt{{\kappa_{ex}}}\left({{\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}}}\right)+2G{e^{i\theta}}{a^{*}},$
(24) $\displaystyle
m(\ddot{x}+{\Gamma_{m}}\dot{x}+\omega_{m}^{2}x)=\hbar\xi{a^{*}}a+\frac{{p_{\phi}^{2}}}{{m{R^{3}}}},$
(25) $\displaystyle\dot{\phi}=\frac{{{p_{\phi}}}}{{m{R^{2}}}},$ (26)
$\displaystyle{{\dot{p}}_{\phi}}=0,$ (27)

where we write the operators for their expectation values by the mean-field
approximation. The steady-state solutions of the system are obtained as

$\begin{split}{\tilde{a}_{s}}&=\frac{{2G{e^{i\theta}}+\kappa-i\tilde{\Delta}}}{{{\kappa^{2}}+{{\tilde{\Delta}}^{2}}-4{G^{2}}}},\\\
{{\tilde{x}}_{s}}&=\frac{{\hbar\xi{{\left|{{\tilde{a}_{s}}}\right|}^{2}}}}{{m\omega_{m}^{2}}}+R{\left({\frac{\Omega}{{{\omega_{m}}}}}\right)^{2}},\end{split}$
(28)

where $\tilde{\Delta}{\rm{=}}{\Delta_{0}}-\xi{{\tilde{x}}_{s}}+{\Delta_{s}}$.
It is worth noting that here, unlike Eq. (11), the intracavity photon number
${\left|{{{\tilde{a}}_{s}}}\right|^{2}}$ and displacement of mechanical
oscillator ${{\tilde{x}}_{s}}$ strongly depend on the magnitude of nonlinear
gain $G$ and phase $\theta$ of the OPA. Eqs. (24)-(27) can be solved
analytically with the linearized ansatz

$\displaystyle a=$
$\displaystyle{\tilde{a}_{s}}+\tilde{A}_{1}^{+}{e^{-i{\Delta_{p}}t}}+\tilde{A}_{1}^{-}{e^{i{\Delta_{p}}t}}+\tilde{A}_{2}^{+}{e^{-2i{\Delta_{p}}t}}+\tilde{A}_{2}^{-}{e^{2i{\Delta_{p}}t}},$
$\displaystyle x=$
$\displaystyle{{\tilde{x}}_{s}}+\tilde{X}_{1}^{+}{e^{-i{\Delta_{p}}t}}+\tilde{X}_{1}^{-}{e^{i{\Delta_{p}}t}}+\tilde{X}_{2}^{+}{e^{-2i{\Delta_{p}}t}}+\tilde{X}_{2}^{-}{e^{2i{\Delta_{p}}t}}.$

After the ansatz, we obtain six algebra equations, which can be divided into
two groups

$\begin{split}{{\tilde{\sigma}}_{1}}\left({{\Delta_{p}}}\right)\tilde{A}_{1}^{+}&=i\xi{{\tilde{a}}_{s}}\tilde{X}_{1}^{+}+2G{e^{i\theta}}\tilde{A}_{1}^{-*}+\sqrt{{\kappa_{ex}}}{\varepsilon_{p}},\\\
{{\tilde{\sigma}}_{2}}\left({{\Delta_{p}}}\right)\tilde{A}_{1}^{-*}&=-i\xi\tilde{a}_{s}^{*}\tilde{X}_{1}^{+}+2G{e^{-i\theta}}\tilde{A}_{1}^{+},\\\
\chi\left({{\Delta_{p}}}\right)\tilde{X}_{1}^{+}&=\hbar\xi({{\tilde{a}}_{s}}\tilde{A}_{1}^{-*}+\tilde{a}_{s}^{*}\tilde{A}_{1}^{+}),\end{split}$
(29)

and

$\begin{split}{{\tilde{\sigma}}_{1}}\left({2{\Delta_{p}}}\right)\tilde{A}_{2}^{+}&=i\xi({{\tilde{a}}_{s}}\tilde{X}_{2}^{+}+\tilde{A}_{1}^{+}\tilde{X}_{1}^{+})+2G{e^{i\theta}}\tilde{A}_{2}^{-*},\\\
{{\tilde{\sigma}}_{2}}\left({2{\Delta_{p}}}\right)\tilde{A}_{2}^{-*}&=-i\xi(\tilde{a}_{s}^{*}\tilde{X}_{2}^{+}+\tilde{A}_{1}^{-*}\tilde{X}_{1}^{+})+2G{e^{-i\theta}}\tilde{A}_{2}^{+},\\\
\chi\left({2{\Delta_{p}}}\right)\tilde{X}_{2}^{+}&=\hbar\xi({{\tilde{a}}_{s}}\tilde{A}_{2}^{-*}+\tilde{a}_{s}^{*}\tilde{A}_{2}^{+}+\tilde{A}_{1}^{+}\tilde{A}_{1}^{-*}),\end{split}$
(30)

with

$\displaystyle{{\tilde{\sigma}}_{1}}\left({n{\Delta_{p}}}\right)$
$\displaystyle=$ $\displaystyle\kappa+i{\tilde{\Delta}}-in{\Delta_{p}},$
$\displaystyle{{\tilde{\sigma}}_{2}}\left({n{\Delta_{p}}}\right)$
$\displaystyle=$ $\displaystyle\kappa-i{\tilde{\Delta}}-in{\Delta_{p}},$
$\displaystyle\chi\left({n{\Delta_{p}}}\right)$ $\displaystyle=$
$\displaystyle m(\omega_{m}^{2}-i{\Gamma_{m}}n{\Delta_{p}}-\Delta_{p}^{2}).$

We get the linear and second-order nonlinear responses of the system

$\begin{split}\tilde{A}_{1}^{+}&=\frac{{\tilde{D}+{{\tilde{\sigma}}_{2}}({\Delta_{p}})\chi({\Delta_{p}})}}{{{{\tilde{f}}_{4}}({\Delta_{p}})+{{\tilde{f}}_{3}}({\Delta_{p}})}}\sqrt{{\kappa_{ex}}}{\varepsilon_{p}},\\\
\tilde{X}_{1}^{+}&=\frac{{\hbar\xi[2G{e^{-i\theta}}{{\tilde{a}}_{s}}+\tilde{a}_{s}^{*}{{\tilde{\sigma}}_{2}}({\Delta_{p}})]}}{{\tilde{D}+{{\tilde{\sigma}}_{2}}({\Delta_{p}})\chi({\Delta_{p}})}}\tilde{A}_{1}^{+},\\\
\tilde{A}_{1}^{-*}&=\frac{{-i\xi\tilde{a}_{s}^{*}}}{{{{\tilde{\sigma}}_{2}}({\Delta_{p}})}}\tilde{X}_{1}^{+}+\frac{{2G{e^{-i\theta}}}}{{{{\tilde{\sigma}}_{2}}({\Delta_{p}})}}\tilde{A}_{1}^{+},\end{split}$
(31)

and

$\begin{split}\tilde{A}_{2}^{+}&=\frac{{i\hbar{\xi^{2}}{{\tilde{f}}_{6}}\tilde{A}_{1}^{+}\tilde{A}_{1}^{-*}+{{\tilde{f}}_{7}}\tilde{A}_{1}^{-*}\tilde{X}_{1}^{+}+i\xi{{\tilde{f}}_{2}}\tilde{A}_{1}^{+}\tilde{X}_{1}^{+}}}{{{{\tilde{f}}_{4}}\left({2{\Delta_{p}}}\right)+{{\tilde{f}}_{3}}\left({2{\Delta_{p}}}\right)}},\\\
\tilde{X}_{2}^{+}&=\frac{{\hbar\xi[{{{\tilde{f}}_{5}}\tilde{A}_{2}^{+}-i\xi{{\tilde{a}}_{s}}\tilde{A}_{1}^{-*}\tilde{X}_{1}^{+}+{{\tilde{\sigma}}_{2}}\left({2{\Delta_{p}}}\right)\tilde{A}_{1}^{+}\tilde{A}_{1}^{-*}}]}}{{{{\tilde{f}}_{2}}}},\\\
\tilde{A}_{2}^{-}&=\frac{{i\xi}}{{{{\tilde{\sigma}}_{2}}{{\left({2{\Delta_{p}}}\right)}^{*}}}}({{\tilde{a}}_{s}}\tilde{X}_{2}^{-}+\tilde{A}_{1}^{-}\tilde{X}_{1}^{-})+\frac{{2G{e^{i\theta}}}}{{{{\tilde{\sigma}}_{2}}{{\left({2{\Delta_{p}}}\right)}^{*}}}}\tilde{A}_{2}^{+*},\\\
\end{split}$ (32)

where

$\begin{split}\tilde{D}&=i\hbar{\xi^{2}}{\left|{{{\tilde{a}}_{s}}}\right|^{2}},\\\
{{\tilde{f}}_{2}}&=\tilde{D}+{{\tilde{\sigma}}_{2}}\left({2{\Delta_{p}}}\right)\chi\left({2{\Delta_{p}}}\right),\\\
{{\tilde{f}}_{3}}\left({n{\Delta_{p}}}\right)&=2i\tilde{D}\Delta+{{\tilde{\sigma}}_{1}}(n{\Delta_{p}}){{\tilde{\sigma}}_{2}}\left({n{\Delta_{p}}}\right)\chi\left({n{\Delta_{p}}}\right),\\\
{{\tilde{f}}_{4}}\left({n{\Delta_{p}}}\right)&=2i\hbar{\xi^{2}}G\left({\tilde{a}_{s}^{*2}{e^{i\theta}}-\tilde{a}_{s}^{2}{e^{-i\theta}}}\right)-4{G^{2}}\chi\left({n{\Delta_{p}}}\right),\\\
{{\tilde{f}}_{5}}&=2G{e^{-i\theta}}{{\tilde{a}}_{s}}+\tilde{a}_{s}^{*}{{\tilde{\sigma}}_{2}}\left({2{\Delta_{p}}}\right),\\\
{{\tilde{f}}_{6}}&=-2G{e^{i\theta}}\tilde{a}_{s}^{*}+\tilde{a}_{s}{{\tilde{\sigma}}_{2}}\left({2{\Delta_{p}}}\right),\\\
{{\tilde{f}}_{7}}&=\hbar{\xi^{3}}\tilde{a}_{s}^{2}-2i\xi
G{e^{i\theta}}\chi\left({2{\Delta_{p}}}\right).\\\ \end{split}$

We obtain the amplitude of the sidebands, which are substituted into the
efficiency of the second-order upper sideband
${{\tilde{\eta}}_{1}}=|-{{\sqrt{{\kappa_{ex}}}\tilde{A}_{2}^{+}}\mathord{/{\vphantom{{\sqrt{{\kappa_{ex}}}\tilde{A}_{2}^{+}}{{\varepsilon_{p}}}}}\kern-1.2pt}{{\varepsilon_{p}}}}|$
and second-order lower sideband
${{\tilde{\eta}}_{2}}=|-\sqrt{{\kappa_{ex}}}\tilde{A}_{2}^{-}/{\varepsilon_{p}}|$.

Figure 9: The efficiency ${{\tilde{\eta}}_{1}}$ of the second-order upper
sideband generation as a function of the probe-pulsed detuning ${\Delta_{p}}$
for different (a) nonlinear gain $G$ under $\theta=0$ and (c) phase $\theta$
of the field driving the OPA under $G=0.2\kappa$. (b) ${{\tilde{\eta}}_{1}}$
varies with ${\Delta_{p}}$ and $G$ under value $\theta=0$. (d)
${{\tilde{\eta}}_{1}}$ varies with ${\Delta_{p}}$ and $\theta$ under
$G=0.2\kappa$. The resonator is stationary ($\Omega=0$). Other parameters are
the same as Fig. 2.

To illustrate the different influences on the second-order sidebands of the
OPA excited by a pump drive of frequency $\omega_{g}=2{\omega_{l}}$, the
efficiency of the second-order upper sideband generation with the resonator
stationary is investigated as a function of frequency
${\Delta_{p}}/{\omega_{m}}$ in Fig. 9. As shown in Fig. 9(a), in the absence
of the OPA, the efficiency ${{\tilde{\eta}}_{1}}$ possesses two near-
symmetrical peaks and a local minimum near the resonance condition
${\Delta_{p}}/{\omega_{m}}=1$. When $G\neq 0$, with the nonlinear gain $G$ of
OPA increasing, the peak of efficiency ${{\tilde{\eta}}_{1}}$ decreases
gradually. But in the driven frequency ${\Delta_{p}}$ range away from the
resonance condition ${\Delta_{p}}={\omega_{m}}$, such as
${\Delta_{p}}>1.01{\omega_{m}}$, the efficiency ${{\tilde{\eta}}_{1}}$ is
enhanced. Moreover, it is noted that the larger the nonlinear gain $G$ of OPA
is, the wider the linewidth of the suppressive windows of the efficiency
${{\tilde{\eta}}_{1}}$ is. Due to the presence of OPA, the suppressive window
will be asymmetric. The result can be applied to determining the excitation
number of atoms and plays important roles in nonlinear media in the optical
properties of the output field. Interestingly, when $G$ increases to
$G=0.8\kappa$, a clear asymmetric linear pattern of the efficiency
${{\tilde{\eta}}_{1}}$ emerges, with a much larger peak at
${\Delta_{p}}=1.01{\omega_{m}}$ than at ${\Delta_{p}}=0.987{\omega_{m}}$. In
Fig. 9(c), we discuss the efficiency ${{\tilde{\eta}}_{1}}$ under different
phase $\theta$ of the field driving the OPA. We find that the phase $\theta$
amplifies the efficiency of the second-order sideband generation, so that the
peak of ${{\tilde{\eta}}_{1}}$ increases from $9.52\%$ to $11.53\%$ for
$\theta=\pi/2$. This is due to the fact that the degenerate parametric
amplifier is a phase-sensitive amplifier, where the phase relationship between
the control laser and signal laser driving the degenerate parametric amplifier
determines the direction of the energy flow, i.e., whether the signal light is
effectively amplified or not. In Fig. 9(b) and (d), ${{\tilde{\eta}}_{1}}$ as
a function of the detuning ${\Delta_{p}}$ and phase $\theta$ of the field
driving the OPA is shown. We can see that the efficiency of the second-order
sideband generation is sensitive to both the nonlinear gain $G$ and phase
$\theta$ changes of the OPA. When
${\Delta_{p}}\in({{\omega_{m}},1.02{\omega_{m}}})$, the influence of the $G$
and $\theta$ on the efficiency ${{\tilde{\eta}}_{1}}$ becomes more obvious.
Specially, as shown in Fig. 9(d), it can be found that at
$\theta\in({0,1.28\pi})$, the efficiency ${{\tilde{\eta}}_{1}}$ is amplified.
When $\theta=0.64\pi$ and ${\Delta_{p}}=1.003{\omega_{m}}$,
${{\tilde{\eta}}_{1}}$ obtains the maximum value $11.73\%$.

Figure 10: The efficiency ${{\tilde{\eta}}_{2}}$ of the second-order lower
sideband generation as a function of the probe-pulsed detuning ${\Delta_{p}}$
for different (a) nonlinear gain $G$ under $\theta=0$ and (c) phase $\theta$
of the field driving the OPA under $G=0.2\kappa$. (b) ${{\tilde{\eta}}_{2}}$
varies with ${\Delta_{p}}$ and $G$ under value $\theta=0$. (d)
${{\tilde{\eta}}_{2}}$ varies with ${\Delta_{p}}$ and $\theta$ under
$G=0.2\kappa$. The resonator is stationary ($\Omega=0$). Other parameters are
the same as Fig. 2.

Next, we discuss the influence of the OPA on the second-order lower sideband
efficiency ${{\tilde{\eta}}_{2}}$. In Fig. 10(a) and (c), we can see that both
$G$ and $\theta$ change the peak of ${{\tilde{\eta}}_{2}}$ (The detailed
results refer to Fig. 10(b) and (d)). As $G$ increases, the position of the
peak shifts to the right, i.e., a larger value of ${\Delta_{p}}$ is needed to
bring ${{\tilde{\eta}}_{2}}$ to its maximum. In particular, when
$G=0.8\kappa$, ${{\tilde{\eta}}_{2}}$ appears as a local minimum at
${\Delta_{p}}=0.993{\omega_{m}}$. As shown in Fig. 10(d),
${{\tilde{\eta}}_{2}}$ is amplified when $\theta\in\left({0,1.14\pi}\right)$,
which obtains the maximum value of $0.95\%$. In general, when the pump laser
frequency driving the OPA is $\omega_{g}=2{\omega_{l}}$, the nonlinear gain
$G$ of the OPA is not significant for the amplification of the second-order
upper and lower sidebands. Compared with the case, where the pump laser
frequency driving OPA is $\omega_{g}={{\omega_{l}}+{\omega_{p}}}$, $G$ can
change the linewidth of the suppressive window of ${{\tilde{\eta}}_{2}}$ and
localization of the sideband efficiency maximum.

Figure 11: The efficiency ${{\tilde{\eta}}_{1}}$ of the second-order upper
sideband generation as a function of the probe-pulsed detuning ${\Delta_{p}}$
for different $G$ with $\theta=0$ at (a) $\Omega=20$ kHz and (b) $\Omega=-20$
kHz. The efficiency ${{\tilde{\eta}}_{1}}$ as a function of the probe-pulsed
detuning ${\Delta_{p}}$ for different $\theta$ with $G=0.2\kappa$ at (c)
$\Omega=20$ kHz and (d) $\Omega=-20$ kHz. Other parameters are the same as
Fig. 2.

As shown in Fig. 11, we discuss the influence of the OPA on the second-order
upper sideband generation when the resonator is rotating. In Fig. 11(a), it
can be seen that when the system is driven from the left-hand side
($\Omega=20$ kHz), the increase of the nonlinear gain $G$ of the OPA enhances
the second-order sideband peak. However, the effect of the OPA in the
transmission window (near ${\Delta_{p}}/{\omega_{m}}=1$) is small, while at
${\Delta_{p}}/{\omega_{m}}<0.996$ and ${\Delta_{p}}/{\omega_{m}}>1.004$, the
OPA has a significant enhancement effect. In Fig. 11(b), we find that when the
system is driven from the right-hand side ($\Omega=-20$ kHz), changing the
nonlinear gain $G$ cannot enhance the second-order sideband peak. But the
increase in the nonlinear gain $G$ of the OPA still makes the linewidth of the
efficiency ${{\tilde{\eta}}_{1}}$ broaden. In Fig. 11(c) and (d),
${{\tilde{\eta}}_{1}}$ as a function of detuning ${\Delta_{p}}$ for the
different $\theta$ at $G=0.2\kappa$ is plotted. In this case, $\Omega=20$ kHz
and $\Omega=-20$ kHz are fixed in Fig. 11(c) and (d), respectively. In detail,
the second-order sideband peak is significantly enhanced when $\theta=\pi$ at
$\Omega=-20$ kHz, but decreased at $\Omega=20$ kHz.

Figure 12: The efficiency ${{\tilde{\eta}}_{1}}$ of the second-order upper
sideband generation as a function of ${\Delta_{p}}$ under different values of
$\Omega$ and incident directions of light, where the nonlinear gain and phase
of the probe field of the OPA are fixed as (a) $G=0,\theta=0$; (b)
$G=0.2\kappa,\theta=0$; (c) $G=0.2\kappa,\theta=3\pi/2$. ${\eta_{1}}$ varies
with ${\Delta_{p}}$ and $\Omega$ under different values (d) $G=0,\theta=0$;
(e) $G=0.2\kappa,\theta=0$; (f) $G=0.2\kappa,\theta=3\pi/2$. These parameters
are the same as Fig. 2.

Figure 13: (a)(b) The efficiency ${{\tilde{\eta}}_{1}}$ of the second-order
upper sideband generation as a function of ${\Delta_{p}}$ under different
values of $\Omega$ and incident directions of light. (c)(d)
${{\tilde{\eta}}_{1}}$ varies with ${\Delta_{p}}$ and $\Omega$. The parameters
chosen are (a)(c) $G=0.4\kappa$, $\theta=\pi/2$ and (b)(d) $G=0.4\kappa$,
$\theta=3\pi/2$. Other parameters are the same as Fig. 2.

In the above discussions, we note that when the frequency $\omega_{g}$ of the
laser field driving the OPA is changed from ${{\omega_{l}}+{\omega_{p}}}$ to
$2{\omega_{l}}$, the influence of the resonator speed, the direction of
incidence of the input fields, the nonlinear gain of the OPA and phase of the
field driving the OPA on the second-order sideband efficiency has a
significant difference in the system. In Figs. 12 and 13, we find in such a
hybrid nonlinear system containing the OPA, the spinning-induced direction-
dependent nonreciprocal behavior remains. We fix the clockwise speed of the
resonator at $20$ kHz and vary the nonlinear gain $G$ and phase $\theta$ of
the field driving the OPA, plotting ${{\tilde{\eta}}_{1}}$ as a function of
${\Delta_{p}}$ and $\Omega$ when the spinning system is driven from the left-
hand side and right-hand side, respectively. In Fig. 12, we choose the same
OPA gain as in Fig. 2 to compare two different OPA cases
($\omega_{g}={{\omega_{l}}+{\omega_{p}}}$ and $\omega_{g}=2{\omega_{l}}$).
When the control laser frequency driving the OPA is
$\omega_{g}=2{\omega_{l}}$, changing the nonlinear gain $G$ can not enhance
the second-order sideband peak. The efficiency of the second-order upper
sideband is not sensitive to the variation of the nonlinear gain of the OPA
and phase of the field driving the OPA. while it is interesting that we can
see with the resonator speed increasing, the second-order sideband peak shifts
to the right regardless of the direction from which the system is driven as
shown in Fig. 12(e) and (f). Furthermore, there are also similarities between
the two different OPA cases, such as compared with the case where the system
is driven from the right side ($\Omega<0$), the influence of resonator
rotation on the second-order sideband enhancement is much more significant
when the system is driven from the left side ($\Omega>0$).

##  VI Nonreciprocal second-order sidebands in non-Markovian systems

When the system interacts with the environment, the dynamics of the system
affected by the environment behaves the dissipation or the backflow
oscillation of the photon from the environment, where the former corresponds
to the Markovian approximation, while the latter exhibits non-Markovian
effects breuer2002 ; breuer1032104012009 ; breuer880210022016 ; Vega015001 .
In Sec.II-Sec.V, we have studied the optomechanical second-order sidebands
under the Markovian approximation. In this section, we investigate the
influences of non-Markovian effects on the efficiency of second-order
sidebands. For this purpose, we consider that the cavity interacts with the
non-Markovian environment consisting of a series of boson modes
(eigenfrequency ${\omega_{k}}$) Xiong2019100 ; Cialdi2019100 ; Tang201297 ;
Groblacher20156 ; Liu20117 ; Hoeppe2012108 ; Xu201082 ; Madsen2011106 ;
Guo2021126 ; Khurana201999 ; Uriri2020101 ; Liu2020102 ; Anderson199347 ;
Li2022129 ; breuer880210022016 ; Vega015001 , where the non-Markovian
environment couples to an external reservoir. In a rotating frame defined by
${{\hat{U}}_{S}}(t)=\exp[-i{\omega_{l}}t({{\hat{a}}^{\dagger}}\hat{a}+\sum\nolimits_{k}{\hat{b}_{k}^{\dagger}}{{\hat{b}}_{k}}+\sum_{j}{\hat{c}_{j}^{\dagger}{{\hat{c}}_{j}}})]$,
the total Hamiltonian (5) is changed to

$\displaystyle{{\hat{H}}_{eff}}=$
$\displaystyle\hbar\left({{\Delta_{0}}-\xi\hat{x}+{\Delta_{s}}}\right){{\hat{a}}^{\dagger}}\hat{a}+\frac{{{{\hat{p}}^{2}}}}{{2m}}+\frac{1}{2}m\omega_{m}^{2}{{\hat{x}}^{2}}$
(33)
$\displaystyle+\frac{{\hat{p}_{\phi}^{2}}}{{2m{{\left({R+\hat{x}}\right)}^{2}}}}+i\hbar
G({\hat{a}^{{\dagger}2}}{e^{-i{\Delta_{p}}t}}{e^{i\theta}}-H.c.)$
$\displaystyle+i\hbar\sqrt{{\kappa_{ex}}}\left[{\left({{\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}}}\right){{\hat{a}}^{\dagger}}-H.c.}\right]$
$\displaystyle+\hbar\sum\limits_{k}{{\Delta_{k}}\hat{b}_{k}^{\dagger}{{\hat{b}}_{k}}}+i\hbar\sum\limits_{k}{({g_{k}}\hat{a}}\hat{b}_{k}^{\dagger}-H.c.)$
$\displaystyle+\hbar\sum\limits_{j}{({{\tilde{\omega}}_{j}}-{\omega_{l}})\hat{c}_{j}^{\dagger}{{\hat{c}}_{j}}}+i\hbar\sum\limits_{jk}{({v_{jk}}{{\hat{c}}_{j}}}\hat{b}_{k}^{\dagger}-H.c.),$

where ${\Delta_{k}}={\omega_{k}}-{\omega_{l}}$ defines the detuning of $k$th
mode (eigenfrequency ${\omega_{k}}$) of the non-Markovian environment from the
driving field. ${{\hat{b}}_{k}}(\hat{b}_{k}^{\dagger})$ is the annihilation
(creation) operator. ${{g_{k}}}$ is the coupling coefficient between cavity
and environment. ${{v_{jk}}}$ denotes the coupling strength between the $k$th
mode of the non-Markovian environment and $j$th mode of the external reservoir
with frequency ${{{\tilde{\omega}}_{j}}}$. ${{{\hat{c}}_{j}}}$ and
$\hat{c}_{j}^{\dagger}$ represent annihilation and creation operators of the
external reservoir, respectively. The dynamics of the system can be derived as

$\displaystyle\frac{d}{{dt}}\hat{a}(t)=-[{\frac{\kappa}{2}+i({{\Delta_{0}}-\xi\hat{x}(t)+{\Delta_{s}}})}]\hat{a}(t)-\sum\limits_{k}{g_{k}^{*}{{\hat{b}}_{k}}(t)}$
$\displaystyle\qquad\qquad+\sqrt{{\kappa_{ex}}}({{\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}}})+2G{{\hat{a}}^{\dagger}}(t){e^{i\theta}}{e^{-i{\Delta_{p}}t}},$
(34)
$\displaystyle\frac{d}{{dt}}{{\hat{b}}_{k}}(t)=-i{\Delta_{k}}{{\hat{b}}_{k}}(t)+{g_{k}}\hat{a}(t)+\sum\limits_{j}{{v_{jk}}{{\hat{c}}_{j}}}(t),$
(35)
$\displaystyle\frac{d}{{dt}}{{\hat{c}}_{j}}(t)=-i({{\tilde{\omega}}_{j}}-{\omega_{l}}){{\hat{c}}_{j}}(t)-\sum\limits_{{k_{1}}}{{v_{j{k_{1}}}^{*}}{{\hat{b}}_{{k_{1}}}}}(t),$
(36)
$\displaystyle\frac{{{d^{2}}}}{{d{t^{2}}}}\hat{x}(t)+{\Gamma_{m}}\frac{d}{{dt}}\hat{x}(t)+\omega_{m}^{2}\hat{x}(t)=\frac{{\hbar\xi}}{m}{{\hat{a}}^{\dagger}}(t)\hat{a}(t)+\frac{{\hat{p}_{\phi}^{2}(t)}}{{{m^{2}}{R^{3}}}},$
(37)
$\displaystyle\frac{d}{{dt}}\hat{\phi}(t)=\frac{{{{\hat{p}}_{\phi}}(t)}}{{m{R^{2}}}},$
(38) $\displaystyle\frac{d}{{dt}}{{\hat{p}}_{\phi}}(t)=0,$ (39)

where the intrinsic loss rate ${\kappa_{a}}=\kappa/2$ is phenomenologically
added in above equations. Eq. (36) gives

$\displaystyle{{\hat{c}}_{j}}(t)=$
$\displaystyle{e^{-i({{\tilde{\omega}}_{j}}-{\omega_{l}})t}}{{\hat{c}}_{j}}(0)$
(40)
$\displaystyle-\sum\limits_{{k_{1}}}{{v_{j{k_{1}}}^{*}}}\int_{0}^{t}{{e^{-i({{\tilde{\omega}}_{j}}-{\omega_{l}})(t-\tau)}}{{\hat{b}}_{{k_{1}}}}(\tau)}d\tau.$

Substituting Eq. (40) into Eq. (35), we get

$\displaystyle\frac{d}{{dt}}{{\hat{b}}_{k}}(t)=$
$\displaystyle-i{\Delta_{k}}{{\hat{b}}_{k}}(t)+{g_{k}}\hat{a}(t)+\sqrt{2\pi}{c_{k,in}}$
(41)
$\displaystyle-\sum\limits_{{k_{1}}}{\int_{0}^{t}{{D_{k{k_{1}}}}(t-\tau)}{{\hat{b}}_{{k_{1}}}}(\tau)d\tau},$

where the input-field operator of the reservoir
${\hat{c}_{k,in}}(t)=\frac{1}{{\sqrt{2\pi}}}\sum_{j}{{v_{jk}}{e^{-i({{\tilde{\omega}}_{j}}-{\omega_{l}})t}}}{{\hat{c}}_{j}}(0)$,
the correlation function
${D_{k{k_{1}}}}(t-\tau)=\sum_{j}{{v_{jk}}v_{j{k_{1}}}^{*}{e^{-i({{\tilde{\omega}}_{j}}-{\omega_{l}})(t-\tau)}}=\int{{{\tilde{J}}_{k{k_{1}}}}(\omega)}}{e^{-i(\omega-{\omega_{l}})(t-\tau)}}d\omega$,
and the spectral density of the reservoir
${{\tilde{J}}_{k{k_{1}}}}(\omega)=\sum_{j}{{v_{jk}}v_{j{k_{1}}}^{*}\delta(\omega-{{\tilde{\omega}}_{j}})}$
with $\delta(\omega)$ being the Dirac delta function. Taking
${{\tilde{J}}_{k{k_{1}}}}(\omega)=\frac{{{\mu_{k}}}}{\pi}{\delta_{k{k_{1}}}}$
($\delta_{k{k_{1}}}$ represents the Kronecker delta symbol, i.e.,
$\delta_{k{k_{1}}}=1$ for $k=k_{1}$, otherwise $\delta_{k{k_{1}}}=0$), and
then ${D_{k{k_{1}}}}(t-\tau)=2{\mu_{k}}\delta(t-\tau){\delta_{k{k_{1}}}}$
breuer2002 ; Gardiner1711022027 , we obtain

$\displaystyle\frac{d}{{dt}}{{\hat{b}}_{k}}(t)=-i{{\tilde{\Delta}}_{k}}{{\hat{b}}_{k}}(t)+{g_{k}}\hat{a}(t)+\sqrt{2\pi}{\hat{c}_{k,in}},$
(42)

with ${{\tilde{\Delta}}_{k}}={\Delta_{k}}-i\mu_{k}$. To simplify the
calculation, we assume $\mu_{k}\equiv\mu$ below, where $\mu$ denotes the decay
from the non-Markovian environment coupling to an external reservoir. The
solution of Eq. (42) is

$\displaystyle{{\hat{b}}_{k}}(t)=$
$\displaystyle{{\hat{b}}_{k}}(0){e^{-i{{\tilde{\Delta}}_{k}}t}}+{g_{k}}\int_{0}^{t}\hat{a}(\tau){e^{-i{{\tilde{\Delta}}_{k}}(t-\tau)}}{d\tau}$
(43)
$\displaystyle+\sqrt{2\pi}\int_{0}^{t}{{\hat{c}}_{k,in}}(\tau){e^{-i{{\tilde{\Delta}}_{k}}(t-\tau)}}{d\tau}.$

The first term on the right-hand side of Eq. (43) represents the freely
propagating parts of the environmental fields and the second term describes
the influence of non-Markovian environment on the cavity. The third term on
the right-hand side of Eq. (43) denotes the influence of the input-field
operator of the reservoir on the non-Markovian environment. Substituting Eq.
(43) into Eq. (34), we obtain an integro-differential equation

$\displaystyle\frac{d}{{dt}}\hat{a}(t)=$
$\displaystyle-[{\frac{\kappa}{2}+i({{\Delta_{0}}-\xi\hat{x}(t)+{\Delta_{s}}})}]\hat{a}(t)$
(44)
$\displaystyle+\sqrt{{\kappa_{ex}}}({{\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}}})+2G{{\hat{a}}^{\dagger}}(t){e^{i\theta}}{e^{-i{\Delta_{p}}t}}$
$\displaystyle+{\hat{K}}(t)+{{\hat{L}}}(t)-\int_{0}^{t}\hat{a}(\tau)f(t-\tau){d\tau},$

where
${\hat{K}}(t)=-\sum_{k}{g_{k}^{*}{{\hat{b}}_{k}}(0){e^{-i{\tilde{\Delta}_{k}}t}}}=\int_{-\infty}^{\infty}{{h^{*}}}(t-\tau){\hat{a}_{in}}(\tau)d\tau$,
$\hat{L}(t)=-\sqrt{2\pi}\sum_{k}{g_{k}^{*}\int_{0}^{t}{{\hat{c}}_{k,in}}(\tau){e^{-i{{\tilde{\Delta}}_{k}}(t-\tau)}}}{d\tau}$,
the input-field operator
${\hat{a}_{in}}(t)=\frac{1}{{\sqrt{2\pi}}}\sum_{k}{{e^{-i{\tilde{\Delta}_{k}}t}}{{\hat{b}}_{k}}(0)}$,
the impulse response function
$h(t)=\frac{{-1}}{{\sqrt{2\pi}}}\sum_{k}{{e^{i{{\tilde{\Delta}}_{k}}t}}{g_{k}}}\equiv\frac{{-1}}{{\sqrt{2\pi}}}\int{{e^{i(\omega-{\omega_{l}})t+\mu
t}}g(\omega)}d\omega$ (we have made the replacement ${g_{k}}\to g(\omega)$ in
the continuum limit), and the correlation function
$f(t)=\sum_{k}{{{\left|{{g_{k}}}\right|}^{2}}{e^{-i{{\tilde{\Delta}}_{k}}t}}}=\int{J(\omega){e^{-i(\omega-{\omega_{l}})t-\mu
t}}d\omega}$ with the spectral density of the non-Markovian environment
$J(\omega)=\sum\nolimits_{k}{|{g_{k}}{|^{2}}}\delta(\omega-{\omega_{k}})$.
Both ${\hat{a}}_{in}(t)$ and ${\hat{c}_{k,in}}$ are the input fields with zero
expectation value ${a}_{in}(t)=\langle{{\hat{a}_{in}}(t)}\rangle=0$ and
${{\rm{c}}_{k,in}}(t)=\langle{{{\hat{c}}_{k,in}}(t)}\rangle=0$ for the
environment and reservoir initialization in the vacuum states, which lead to
${{K}}(t)=\langle{\hat{K}}(t)\rangle=0$ and
${{L}}(t)=\langle{\hat{L}}(t)\rangle=0$. We define the spectral response
function as

$\displaystyle
g(\omega)=\sqrt{\frac{{{\kappa_{ex}}}}{{2\pi}}}\frac{{{\lambda_{1}}}}{{{\lambda_{1}}-i(\omega-{\omega_{l}})}},$
(45)

where ${\lambda_{1}}$ is the environmental spectrum width and
${\kappa_{ex}}=\kappa$ is the cavity dissipation through the input and output
ports. The spectral density of the environment is shen880338352013 ;
zhang870321172013 ; diosi850341012012 ; xiong860321072012 ; shen950121562017

$\displaystyle\begin{aligned}
J(\omega)=\frac{{{\kappa_{ex}}}}{{2\pi}}\frac{{\lambda_{1}^{2}}}{{\lambda_{1}^{2}+{(\omega-\omega_{l})^{2}}}},\end{aligned}$
(46)

which corresponds to the Lorentzian spectral density. With Eqs. (45) and (46),
we get
$h(\tau-t)=-\sqrt{{\kappa_{ex}}}{\lambda_{1}}{e^{-{({\lambda_{1}}+i\mu)}(t-\tau)}}\theta(t-\tau)$
and
$f(t-\tau)=\frac{1}{2}{\kappa_{ex}}{\lambda_{1}}{e^{-{({\lambda_{1}}+i\mu)}\left|{t-\tau}\right|}}$,
where $\theta(t-t^{\prime})$ is the unit step function,
$\theta(t-t^{\prime})=1$ for $t\geq t^{\prime}$, which represents a Gaussian
Ornstein-Uhlenbeck process uhlenbeck368231930 ; gillespie5420841996 ;
jing1052404032010 .

Figure 14: (a)(b) The efficiency ${\eta^{\prime}_{1}}$ of the second-order
upper sideband generation as a function of ${\Delta_{p}}$, which corresponds
to the Markovian and non-Markovian environments with the different
environmental spectrum width ${\lambda_{1}}$ without the OPA involvement
$(G=0)$. (c)(d)(e)(f) ${\eta^{\prime}_{1}}$ varies with ${\Delta_{p}}$ and
${\lambda_{1}}$. The rotation speed is set as (a)(c)(e) $\Omega=0$ and
(b)(d)(f) $\Omega=7.7$ kHz. The parameter $\mu$ denotes the decay from the
non-Markovian environment coupling to an external reservoir, where $\mu=0$ for
(c) and (d), while $\mu=5\omega_{m}$ for (e) and (f). Other parameters are the
same as Fig. 2.

For convenience, we take the expectation values of the operator equations by
defining $a\equiv\langle{\hat{a}}\rangle$ , $x\equiv\langle{\hat{x}}\rangle$ ,
$\phi\equiv\langle{\hat{\phi}}\rangle$ and
${p_{\phi}}\equiv\langle{{{\hat{p}}_{\phi}}}\rangle$. The steady-state
solution of the non-Markovian system can be obtained from Eq. (44) as

$\displaystyle{a^{\prime}_{s}}=$
$\displaystyle\frac{{\sqrt{{\kappa_{ex}}}{\varepsilon_{l}}}}{{\kappa+i{\Delta^{\prime}}}},$
(47) $\displaystyle{x^{\prime}_{s}}=$
$\displaystyle\frac{{\hbar\xi{{|{a^{\prime}_{s}}|}^{2}}}}{{m\omega_{m}^{2}}}+R{\left({\frac{\Omega}{{{\omega_{m}}}}}\right)^{2}},$

where $\Delta^{\prime}{\rm{=}}{\Delta_{0}}-\xi{x^{\prime}_{s}}+{\Delta_{s}}$.
We make the ansatz

$\displaystyle a=$
$\displaystyle{a^{\prime}_{s}}+{{A^{\prime}}_{1}^{+}}{e^{-i{\Delta_{p}}t}}+{{A^{\prime}}_{1}^{-}}{e^{i{\Delta_{p}}t}}+{{A^{\prime}}_{2}^{+}}{e^{-2i{\Delta_{p}}t}}+{{A^{\prime}}_{2}^{-}}{e^{2i{\Delta_{p}}t}},$
$\displaystyle x=$
$\displaystyle{x^{\prime}_{s}}+{{X^{\prime}}_{1}^{+}}{e^{-i{\Delta_{p}}t}}+{{X^{\prime}}_{1}^{-}}{e^{i{\Delta_{p}}t}}+{{X^{\prime}}_{2}^{+}}{e^{-2i{\Delta_{p}}t}}+{{X^{\prime}}_{2}^{-}}{e^{2i{\Delta_{p}}t}},$

We get the linear response of the probe field

$\displaystyle{\sigma^{\prime}_{1}}({{\Delta_{p}}}){{A^{\prime}}_{1}^{+}}$
$\displaystyle=\Lambda({{\Delta_{p}}})[i\xi{a^{\prime}_{s}}{{X^{\prime}}_{1}^{+}}+2G{e^{i\theta}}{{a^{\prime}}_{s}^{*}}+\sqrt{{\kappa_{ex}}}{\varepsilon_{p}}],$
(48)
$\displaystyle{\sigma^{\prime}_{2}}({{\Delta_{p}}}){{A^{\prime}}_{1}^{-*}}$
$\displaystyle=-i\xi\Lambda({{\Delta_{p}}}){{a^{\prime}}_{s}^{*}}{{A^{\prime}}_{1}^{+}},$
$\displaystyle\chi({{\Delta_{p}}}){{X^{\prime}}_{1}^{+}}$
$\displaystyle=\hbar\xi({a^{\prime}_{s}}{{A^{\prime}}_{1}^{-*}}+{{a^{\prime}}_{s}^{*}}{{A^{\prime}}_{1}^{+}}),$

and second-order sideband process

$\displaystyle{\sigma^{\prime}_{1}}(2{\Delta_{p}}){{A^{\prime}}_{2}^{+}}$
$\displaystyle=\Lambda({2{\Delta_{p}}})[i\xi({a^{\prime}_{s}}{{X^{\prime}}_{2}^{+}}+{{A^{\prime}}_{1}^{+}}{{X^{\prime}}_{1}^{+}})+2G{e^{i\theta}}{{A^{\prime}}_{1}^{-*}}],$
(49)
$\displaystyle{\sigma^{\prime}_{2}}({2{\Delta_{p}}}){{A^{\prime}}_{2}^{-*}}$
$\displaystyle=-i\xi\Lambda({2{\Delta_{p}}})({{a^{\prime}}_{s}^{*}}{{X^{\prime}}_{2}^{+}}+{{A^{\prime}}_{1}^{-*}}{{X^{\prime}}_{1}^{+}}),$
$\displaystyle\chi({2{\Delta_{p}}}){{X^{\prime}}_{2}^{+}}$
$\displaystyle=\hbar\xi({{a^{\prime}}_{s}^{*}}{{A^{\prime}}_{2}^{+}}+{a^{\prime}_{s}}{{A^{\prime}}_{2}^{-*}}+{{A^{\prime}}_{1}^{-*}}{{A^{\prime}}_{1}^{+}}),$

with

$\displaystyle\Lambda\left({n{\Delta_{p}}}\right)$ $\displaystyle=$
$\displaystyle{{\lambda_{1}}+i\mu}-in{\Delta_{p}},$
$\displaystyle{\sigma^{\prime}_{1}}\left({n{\Delta_{p}}}\right)$
$\displaystyle=$ $\displaystyle\kappa{\lambda_{1}}-\frac{{i\kappa
n{\Delta_{p}}}}{2}+\Lambda({n{\Delta_{p}}})(i\Delta-in{\Delta_{p}}),$
$\displaystyle{\sigma^{\prime}_{2}}\left({n{\Delta_{p}}}\right)$
$\displaystyle=$ $\displaystyle\kappa{\lambda_{1}}-\frac{{i\kappa
n{\Delta_{p}}}}{2}-\Lambda({n{\Delta_{p}}})(i\Delta+in{\Delta_{p}}),$
$\displaystyle\chi\left({n{\Delta_{p}}}\right)$ $\displaystyle=$
$\displaystyle m(\omega_{m}^{2}-i{\Gamma_{m}}n{\Delta_{p}}-\Delta_{p}^{2}).$

Through the derived non-Markovian input-output relation by Eq. (44), we obtain
the expected value of the output field

${a_{out}}(t)={a_{in}}(t)+\int_{0}^{t}{h(\tau-t)}a(\tau)d\tau.$ (51)

Thus in the non-Markovian case, the efficiency of second-order upper sideband
is defined as

${\eta^{\prime}_{1}}=\left|{-\frac{{\sqrt{{\kappa_{ex}}}{\lambda_{1}}{{A^{\prime}}_{2}^{+}}\frac{1}{{{{\lambda_{1}}+i\mu}-2i{\Delta_{p}}}}}}{{{\varepsilon_{p}}}}}\right|.$
(52)

With Eq. (52), we consider two cases (i) and (ii) separately.

(i) In the first case, we take the decay $\mu=0$ in Eq. (52). In Fig. 14(a)
with the decay $\mu=0$, resonator stationary but without the participation of
the OPA, we show the efficiency of second-order upper sideband generation as a
function of ${\Delta_{p}}$ with the different spectral width of environment
${\lambda_{1}}$. For a given spectral width of environment, decreasing from
${\lambda_{1}}=10{\omega_{m}}\sim 2{\omega_{m}}$, we find from the figure that
the second-order upper sideband ${\eta^{\prime}_{1}}$ gradually decreases,
whose two located peaks become increasingly asymmetric in the non-Markovian
environment. Interestingly, from Fig. 14(b) with the decay $\mu=0$, when the
light comes from the right side and $\Omega=7.7$ kHz, ${\eta^{\prime}_{1}}$
becomes symmetric in the non-Markovian environment at
${\lambda_{1}}=2{\omega_{m}}$. That is, by controlling the rotation speed of
the resonator and incident direction of the input fields, the symmetry of the
second-order sideband is restored, but with a change in height compared with
the Markovian environment. With the purpose of seeing the influence of the
environmental spectrum width on the second-order sideband generation more
clearly, the efficiency ${\eta^{\prime}_{1}}$ as a function of both
${\Delta_{p}}$ and ${\lambda_{1}}$ is shown in Fig. 14(c) and (d) with the
decay $\mu=0$.

Figure 15: The efficiency ${\eta^{\prime}_{1}}$ of the second-order upper
sideband generation as a function of ${\Delta_{p}}$ under different values of
$\Omega$ and incident directions of light, where we take $G=0$. This figure
shows the consistency of nonreciprocal second-order sidebands between non-
Markovian limit with ${\lambda_{1}}=200{\omega_{m}}$ and Markovian
approximation. Other parameters are the same as Fig. 2.

As the spectrum width of the environment is further increased, the efficiency
of second-order upper sideband generation increases. For the sake of clarity,
we separately draw the non-Markovin case and the Markovian limit case where
the environmental spectrum width ${\lambda_{1}}=200{\omega_{m}}$ for the
condition that the resonator is stationary and no OPA is involved in Fig. 15
with the decay $\mu=0$. This figure shows the consistency of nonreciprocal
second-order upper sideband between non-Markovian limit with
${\lambda_{1}}=200{\omega_{m}}$ and Markovian approximation, regardless of the
incident direction of the input fields. This originates from the fact that the
correlation function $f(t)$ and impulse response function $h(t)$ tend to
$\kappa_{ex}\delta(t)$ and $-\sqrt{\kappa_{ex}}\delta(t)$ in the wideband
limit (i.e., the spectrum width $\lambda_{1}$ approaches infinity),
respectively, which leads to Eqs. (44) and (51) in the non-Markovian regime
returning back to Eqs. (6) and (16) under the Markovian approximation.

Figure 16: (a)(b) The efficiency ${\eta^{\prime}_{1}}$ of the second-order
upper sideband generation as a function of ${\Delta_{p}}$ under different
values of $\Omega$ and incident directions of light in the non-Markovian
environment and without the participation of the OPA $(G=0)$. (c)(d)(e)(f)
${\eta^{\prime}_{1}}$ varies with ${\Delta_{p}}$ and $\Omega$. The
environmental spectrum widths are (a)(c)(e) ${\lambda_{1}}=0.5{\omega_{m}}$
and (b)(d)(f) ${\lambda_{1}}=2{\omega_{m}}$, respectively. (c) and (d) take
the decay $\mu=0$, while the decay $\mu=5\omega_{m}$ corresponds to (e) and
(f). Other parameters are the same as Fig. 2.

Fig. 16(a)-(d) with the decay $\mu=0$ shows the spinning-induced direction-
dependent nonreciprocal behavior of second-order upper sideband in the non-
Markovian environment but without the participation of the OPA. We note that
on the one hand, the efficiency of second-order sideband ${\eta^{\prime}_{1}}$
is very sensitive to the environmental spectrum width. On the other hand, the
operating bandwidth for observing an obvious nonreciprocal enhancement of
second-order sideband changes in the non-Markovian environment. Compared with
the Markovian environment in Fig. 15 with the decay $\mu=0$, the operating
bandwidth becomes significantly wider at frequency ${\Delta_{p}}>{\omega_{m}}$
and narrower at ${\Delta_{p}}<{\omega_{m}}$.

Figure 17: The efficiency ${\eta^{\prime}_{1}}$ of the second-order upper
sideband generation as a function of ${\Delta_{p}}$ for different nonlinear
gain $G$ of the OPA, where $\theta=0$, $\Omega=0$, and the decay $\mu=0$. The
environmental spectrum widths are (a) ${\lambda_{1}}=0.5{\omega_{m}}$, (b)
${\lambda_{1}}=2{\omega_{m}}$, (c) ${\lambda_{1}}=5{\omega_{m}}$, and (d)
${\lambda_{1}}=30{\omega_{m}}$, respectively. Other parameters are the same as
Fig. 2.

Figs. 14, 15 and 16 with the decay $\mu=0$ present the influence of pure non-
Markovian effect on the second-order sideband without the participation of the
OPA ($G=0$). In Fig. 17 with the decay $\mu=0$, we show the variation of
second-order upper sideband efficiency in the presence of both non-Markovian
effect and OPA. As expected, when the nonlinear gain $G$ of the OPA increases
from $0$ to $0.6\kappa$, the efficiency ${\eta^{\prime}_{1}}$ is significantly
enhanced. Moreover, the non-Markovian effect is more pronounced for
${\eta^{\prime}_{1}}$ when the environmental spectrum width is small (i.e.,
${\lambda_{1}}<2{\omega_{m}}$). As shown in Fig. 17(d) with the decay $\mu=0$
at ${\lambda_{1}}=30{\omega_{m}}$, the enhancement effect of the OPA for
second-order sideband is almost identical to the case of Markovian limit.

(ii) In the second case, we take the decay $\mu=5\omega_{m}$ in Eq. (52). The
influences of the decay from the non-Markovian environment coupling to an
external reservoir on the efficiency of second-order upper sidebands are shown
in Figs. 14, 15 and 16 with $\mu=5\omega_{m}$. We find that the decay $\mu$
has large influences on the efficiency of second-order upper sidebands in non-
Markovian regimes, while it has almost no influence on the efficiency of
second-order upper sidebands under the Markovian approximation. This is
because the decay $\mu$ is comparable to the spectral width $\lambda_{1}$ of
the non-Markovian environment revealed from Eqs. (LABEL:second3xxc) and (52)
(see Fig. 14(a)(b)(e)(f) and Fig. 16(a)(b)(e)(f)) since the spectral width
$\lambda_{1}$ takes finite values in non-Markovian regimes. However, the
spectral width $\lambda_{1}$ tends to infinity (i.e., $\lambda_{1}\to\infty$)
under the Markovian approximation, which leads to that the decay $\mu$ is
negligible compared with the spectral width $\lambda_{1}$ due to
$\mu\ll\infty$ in Eqs. (LABEL:second3xxc) and (52) (see Fig. 14(a)(b) and Fig.
15).

##  VII Conclusion

In summary, we have theoretically studied the second-order OMIT sidebands and
group delays in a spinning resonator containing an optical parametric
amplifier. We discuss the influence of the OPA driven by different pumping
frequencies on the second-order sideband generation. The results show that the
second-order sidebands in the rotating resonator can be greatly enhanced in
the presence of the OPA and still remain the nonreciprocal behavior due to the
optical Sagnac effect. The second-order sidebands can be adjusted
simultaneously by the pumping frequency and phase of the field driving the
OPA, the gain coefficient of the OPA, the rotation speed of the resonator, and
the incident direction of the input fields. When the OPA is excited by a pump
driving with the frequency $\omega_{g}={{\omega_{l}}+{\omega_{p}}}$, the
higher nonlinear gain of the OPA is, the stronger the second-order sidebands
are. At this point, the OPA can only enhance the second-order sidebands but
cannot change the position of the peaks and the non-reciprocal nature due to
resonator rotation, which maintains the localization of the maximum value of
the sideband efficiency. When the OPA is excited by a pump driving with the
frequency $\omega_{g}=2{\omega_{l}}$, the nonlinear gain of the OPA cannot
enhance the second-order sidebands, which can only be achieved by adjusting
the phase of the field driving the OPA. The OPA can also change the linewidth
of the suppressive window of the second-order sidebands, which can be applied
to determining the excitation number of atoms and plays important roles in
nonlinear media in the optical properties of the output field. Combining the
Sagnac transformation and the presence of the OPA, we demonstrate that the
group delay of the second-order upper sideband can be tuned by adjusting the
nonlinear gain and phase of the field driving the OPA, the rotation speed of
the resonator and incident direction of the input fields, which allows us to
realize a tunable switch from slow light to fast light in the spinning
optomechanical system. Moreover, we extend the study of second-order sidebands
from the Markovian to the non-Markovian bath, which consists of a collection
of infinite oscillators (bosonic photonic modes). We find the second-order
OMIT sidebands in a spinning resonator exhibit a transition from the non-
Markovian to Markovian regime by controlling environmental spectral width.
Finally, we investigate the influences of the decay from the non-Markovian
environment coupling to an external reservoir on the efficiency of second-
order upper sidebands.

These results indicate the advantage of using a hybrid nonlinear system and
contribute to a better understanding of light propagation in nonlinear
optomechanical devices, which provides potential applications for precision
measurement, optical communications, and quantum sensing. Expansions of the
above non-Markovian nonreciprocal second-order sidebands to various general
nonlinear physical models, e.g., (1) ${\chi^{(2)}}$ nonlinear materials
${{\hat{a}}^{2}}{{\hat{b}}^{\dagger}}+\hat{b}{{\hat{a}}^{{\dagger}2}}$ zz1 ;
zz2 , (2) Kerr nonlinear mediums ${{\hat{a}}^{{\dagger}2}}{{\hat{a}}^{2}}$ zz3
; zz4 , and (3) quadratic optomechanical systems
${{\hat{a}}^{\dagger}}\hat{a}(\hat{b}+{{\hat{b}}^{\dagger}})^{2}$
Aspelmeyer861391 ; Thompson45272 ; jack630438032001zs ; jack630438032001zzz ;
jack630438032001z1s ; jack630438032001zs3 , deserve future investigations.

##  VIII ACKNOWLEDGMENTS

This work was supported by National Natural Science Foundation of China under
Grants No. 12274064, Scientific Research Project for Department of Education
of Jilin Province under Grant No. JJKH20190262KJ, and Natural Science
Foundation of Jilin Province (subject arrangement project) under Grant No.
20210101406JC.

##

*

##  Appendix A Derivation of Eqs. (6)-(9)

In order to give the origin of $\Gamma_{m}$ in Eq. (7), we add the coupling
Hamiltonian $\hat{H}_{CL}$ Weiss1999 ; Caldeira1983347 ; Caldeira5871983 ;
Grabert1151988 ; Spiechowicz052107 ; Einsiedler0222282020 ; Sinha051111 ;
Sun5118451995  between the mechanical mode and a Bosonic bath consisting of a
set of harmonic oscillators with mass $m_{l}$ and frequency $\Omega_{l}$ to
Eq. (5) as follows

$\displaystyle\hat{H}_{CL}=\sum\limits_{l}{\left[{\frac{{\hat{P}_{l}^{2}}}{{2{M_{l}}}}+\frac{{{M_{l}}\Omega_{l}^{2}}}{2}{{\left({{{\hat{C}}_{l}}-\frac{{{v_{l}}}}{{{M_{l}}\Omega_{l}^{2}}}\hat{x}}\right)}^{2}}}\right]},$
(53)

where $\hat{C}_{l}$ and $\hat{P}_{l}$ are the coordinate and momentum of the
harmonic oscillators, respectively, while $v_{l}$ denotes coupling strength
between mechanical mode and bath. The counterterm proportional to
$\hat{x}^{2}$ is typically introduced in the Hamiltonian, which accounts for a
renormalization of the central oscillator frequency due to the interaction
with the bath Weiss1999 ; Caldeira1983347 ; Caldeira5871983 ; Grabert1151988 ;
Spiechowicz052107 ; Einsiedler0222282020 ; Sinha051111 ; Sun5118451995 . The
Heisenberg equations read

$\displaystyle\frac{d}{{dt}}\hat{a}$ $\displaystyle=$
$\displaystyle-\left[{\kappa+i\left({{\Delta_{0}}-\xi
x+{\Delta_{s}}}\right)}\right]\hat{a}$
$\displaystyle+\sqrt{{\kappa_{ex}}}({\varepsilon_{l}}+{\varepsilon_{p}}{e^{-i{\Delta_{p}}t}})+2G{{\hat{a}}^{\dagger}}{e^{i\theta}}{e^{-i{\Delta_{p}}t}},$
$\displaystyle\frac{d}{{dt}}\hat{x}$ $\displaystyle=$
$\displaystyle\frac{{\hat{p}}}{m},$ (55) $\displaystyle\frac{d}{{dt}}\hat{p}$
$\displaystyle=$
$\displaystyle-m\omega_{m}^{2}\hat{x}+\sum\limits_{l}{{v_{l}}{{\hat{C}}_{l}}}-\sum\limits_{l}{\frac{{v_{l}^{2}}}{{{M_{l}}\Omega_{l}^{2}}}}\hat{x}$
(56)
$\displaystyle+\hbar\xi{{\hat{a}}^{\dagger}}\hat{a}+\frac{{\hat{p}_{\phi}^{2}}}{{m{R^{3}}}},$
$\displaystyle\frac{d}{{dt}}{{\hat{C}}_{l}}$ $\displaystyle=$
$\displaystyle\frac{{{{\hat{P}}_{l}}}}{{{M_{l}}}},$ (57)
$\displaystyle\frac{d}{{dt}}{{\hat{P}}_{l}}$ $\displaystyle=$
$\displaystyle-{M_{l}}\Omega_{l}^{2}{{\hat{C}}_{l}}+{v_{l}}\hat{x},$ (58)
$\displaystyle\frac{d}{{dt}}\hat{\phi}$ $\displaystyle=$
$\displaystyle\frac{{{\hat{p}_{\phi}}}}{{m{R^{2}}}},$ (59)
$\displaystyle\frac{d}{{dt}}\hat{{p}}_{\phi}$ $\displaystyle=$ $\displaystyle
0,$ (60)

where the Heisenberg operator ${\hat{x}}(t)$ is abbreviated as
$\hat{x}\equiv\hat{x}(t)={e^{i{{\hat{H}}_{T}}t/\hbar}}\hat{x}(0){e^{i{{\hat{H}}_{T}}t/\hbar}}$
with ${{{\hat{H}}_{T}}}={{\hat{H}}_{eff}}+\hat{H}_{CL}$ (${\hat{H}}_{eff}$ is
given by Eq. (5)), and the other operators also have similar expressions. Eqs.
(6)(8)(9) are consistent with Eqs. (LABEL:Heisenbergz123)(59)(60),
respectively. Differentiating Eqs. (55) and (57), together with Eqs. (56) and
(58), we have

$\displaystyle
m\left[{\frac{{{d^{2}}}}{{d{t^{2}}}}\hat{x}+\omega_{m}^{2}\hat{x}}\right]=\sum\limits_{l}{{v_{l}}{{\hat{C}}_{l}}}-\sum\limits_{l}{\frac{{v_{l}^{2}}}{{{M_{l}}\Omega_{l}^{2}}}}\hat{x}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\hbar\xi{{\hat{a}}^{\dagger}}\hat{a}+\frac{{\hat{p}_{\phi}^{2}}}{{m{R^{3}}}},$
(61)
$\displaystyle\frac{{{d^{2}}}}{{d{t^{2}}}}{{\hat{C}}_{l}}+\Omega_{l}^{2}{{\hat{C}}_{l}}=\frac{{{v_{l}}}}{{{M_{l}}}}\hat{x}.$
(62)

The solution of Eq. (62) is

$\displaystyle{{\hat{C}}_{l}}=$
$\displaystyle{{\hat{C}}_{l}}(0)\cos{\Omega_{l}}t+\frac{{{{\hat{P}}_{l}}(0)}}{{{M_{l}}{\Omega_{l}}}}\sin{\Omega_{l}}t$
(63)
$\displaystyle+{v_{l}}\int_{0}^{t}{\frac{{\sin{\Omega_{l}}(t-\tau)}}{{{M_{l}}{\Omega_{l}}}}}\hat{x}(\tau)d\tau.$

Substituting Eq. (63) into Eq. (61) gives

$\displaystyle
m\left[{\frac{{{d^{2}}}}{{d{t^{2}}}}\hat{x}+\omega_{m}^{2}\hat{x}+\int_{0}^{t}{\eta(t-\tau)}\hat{x}(\tau)d\tau}\right]+\sum\limits_{l}{\frac{{v_{l}^{2}}}{{{M_{l}}\Omega_{l}^{2}}}}\hat{x}$
(64)
$\displaystyle=\hat{F}(t)+\hbar\xi{{\hat{a}}^{\dagger}}\hat{a}+\frac{{\hat{p}_{\phi}^{2}}}{{m{R^{3}}}},$

with
$\hat{F}(t)=\sum\nolimits_{l}{{v_{l}}}[{{\hat{C}}_{l}}(0)\cos{\Omega_{l}}t+({{\hat{P}}_{l}}(0)/{M_{l}}{\Omega_{l}})\sin{\Omega_{l}}t]$.
The kernel $\eta(t)$ equals $\frac{{d{\alpha}(t)}}{{dt}}$, where the
correlation function
${\alpha}(t)=\sum\nolimits_{l}{v_{l}^{2}\cos{\Omega_{l}}t/(m{M_{l}}\Omega_{l}^{2})}\equiv\int{I(\omega)}\cos(\omega)d\omega$
with the spectral density
$I(\omega)=\sum\nolimits_{l}{\frac{{v_{l}^{2}}}{{m{M_{l}}\Omega_{l}^{2}}}\delta(\omega-{\Omega_{l}})}$.
Taking expectation values (The states of each part for the system are
initially prepared in their respective vacuum states) to Eq. (64) leads to

$\displaystyle
m\left[{\frac{{{d^{2}}}}{{d{t^{2}}}}x+\omega_{m}^{2}x+\int_{0}^{t}{\eta(t-\tau)}x(\tau)d\tau}\right]+\sum\limits_{l}{\frac{{v_{l}^{2}}}{{{M_{l}}\Omega_{l}^{2}}}}x$
(65) $\displaystyle=\hbar\xi{a^{*}}a+\frac{{p_{\phi}^{2}}}{{m{R^{3}}}},$

where we have used the expectation value
${{F(t)}}=\langle{{{\hat{F}(t)}}}\rangle$ of $\hat{F}(t)$ equalling zero. With
the partial integration and $x(0)=0$ (the expectation value of $\hat{x}(0)$ is
${{x(0)}}=\langle{{{\hat{x}(0)}}}\rangle$), Eq. (65) is reduced as

$\displaystyle
m[\ddot{x}+\int_{0}^{t}{\alpha(t-\tau)}\dot{x}(\tau)d\tau+\omega_{m}^{2}x]=\hbar\xi{a^{*}}a+\frac{{p_{\phi}^{2}}}{{m{R^{3}}}}.$
(66)

With the Lorentzian spectral density
$I(\omega)={\Gamma_{m}}{\Lambda^{2}}/[\pi({\omega^{2}}+{\Lambda^{2}})]$
breuer2002 ; breuer1032104012009 ; breuer880210022016 ; Vega015001 , we obtain
$\alpha(t)={\Gamma_{m}}\Lambda{e^{-\Lambda|t|}}$, where the parameter
$\Lambda$ defines the spectral width of the bath, which is connected to the
bath correlation time $T_{B}$ by the relation $T_{B}=\Lambda^{-1}$, while the
time scale for the state of the system changing is given by
$T_{S}=\Gamma_{m}^{-1}$. Under the Markovian approximation
($\Lambda\to\infty$), we get

$\displaystyle\alpha(t)\to 2\Gamma_{m}\delta(t).$ (67)

Eq. (7) can be obtained by substituting Eq. (67) into Eq. (66), where we have
used the identity
$\int_{0}^{t}{\delta(t-\tau)}\dot{x}(\tau)d\tau=\frac{1}{2}\dot{x}(t)$
Gardiner1711022027 .

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