[orcid=0000-0002-1346-0167]

[1]

1]organization=Fujian Key Laboratory of Light Propagation and Transformation &
Institute of Systems Science, College of Information Science and Engineering,
Huaqiao University, addressline=, city=Xiamen, postcode=361201, country=China

[] [1]

[]

[orcid=0000-0002-7169-0446] [2]

2]organization=Center for Quantum Optics and Quantum Information, Universidad
Mayor, addressline=Camino La Pirámide 5750, city=Huechuraba, state=RM,
country=Chile

[cor1]These authors contribute equally to this work [cor2]corresponding author

# Nonlinear optomechanical resonance entering a self-organized energy transfer
pattern

Qing Lin  [  Yi Wu  Gang Li  Bing He<EMAIL_ADDRESS>[

###### Abstract

The energy transfer between different subsystems or different vibration modes
is always one of the most interested problems in the study of the resonance
phenomena in coupled nonlinear dynamical systems. With an optomechanical
system operating in the regime of unresolved sideband, where its mechanical
frequency is lower than the cavity field damping rate, we illustrate the
existence of a special nonlinear resonance phenomenon. This type of previously
unknown resonance manifests an organized pattern of the coupled cavity field
and mechanical oscillation, so that the cavity field precisely pushes the
mechanical oscillator within an appropriate small time window in each
mechanical oscillation period and the mechanical energy will increase by a
jump of almost fixed amount after each oscillation cycle. The scenario is
realized at a resonance point where the frequency difference of two driving
fields matches the mechanical frequency of the system, and this condition of
drive-frequency match is found to trigger a mechanism to lock the two
subsystems of an unresolved-sideband optomechanical system into a highly
ordered energy transfer as the above mentioned. Due to a significantly
enhanced nonlinearity in the vicinity of the resonance point, optical
frequency combs can be generated under pump powers of thousand times lower, as
compared to the use of a single-tone driving field for the purpose. An
unresolved sideband system under the drives without satisfying the resonance
condition also demonstrates other interesting dynamical behaviors. Most of
all, by providing a realistic picture for the nonlinear optomechanical
dynamics in unresolved sideband regime, our study points to a direction to
observe novel dynamical phenomena and realize other applications with the
systems of less technical restrictions.

###### keywords:

nonlinear resonance optomechanics unresolved sideband regime dynamical pattern
locking

##  1 Introduction

Resonance phenomena are ubiquitous in both nature and artificial structures,
and refer to the maximum response of a vibration to external oscillatory
forces. Beyond the scenarios of a single oscillator, such as the well-known
stochastic resonance [1] and parametric resonance [2], the nonlinear resonance
phenomena in coupled systems were less studied but are more important to
applications. For example, internal resonance in coupled structures [3, 4, 5,
6] is highly interested and useful for its energy transfer between different
vibration modes under the condition that their frequencies are in some proper
ratios. The current work is about a nonlinear resonance in optomechanical
systems (OMS), which manifests a unique pattern of energy transfer between
coupled cavity field and mechanical oscillator.

In addition to their potential values in quantum technology [7, 8], OMSs
provide a good platform to study nonlinear dynamics. Nonlinear dynamical
behaviors in OMS were previously known as the bifurcations into limit cycles
[9, 10, 11, 12, 13, 14, 15, 16] and chaos [17, 18, 19, 20, 21]. The dynamical
scenarios in the first category can be summarized by the response of an OMS to
the external driving fields of varied frequencies and intensities. In the
regime where the systems’ mechanical frequencies $\omega_{m}$ exceed their
optical cavity damping rates $\kappa$, a driving laser will turn an OMS from a
steady state to an oscillation when its power is above a threshold of Hopf
bifurcation. Direct numerical calculations based on the full nonlinear
dynamics [22] show that, along the way of scanning the frequency of a driving
laser of the same power, the induced mechanical oscillation has the largest
amplitude at where the drive frequency is blue-detuned from the system’s
resonant cavity frequency exactly by the mechanical frequency $\omega_{m}$.
This is a resonance point of the same dynamical pattern of harmonic
oscillation, commonly seen in nonlinear dynamical systems. Parallel to this
category of nonlinear dynamics, most recent researches on the regime
$\omega_{m}/\kappa>1$ concern the side of quantum mechanics, such as
mechanical ground states (see, e.g. [23, 24, 25]) and others (see, e.g. [26,
27]), which can be well described by the linearized dynamics around a steady
state [7] or by another quantum dynamical approach [22, 28]. To demonstrate
quantum properties, an OMS should be ideally fabricated to meet the resolved
sideband condition $\omega_{m}\gg\kappa$, and this requirement poses a
technical challenge of tremendously increasing the optical finesse and limits
the mass/size of the mechanical element of an OMS. Instead, the phenomena
described in the current work must exist in another regime where
$\omega_{m}<\kappa$. Most previous experiments in such unresolved sideband
regime [29, 30, 31, 32] were performed in the bad cavity limit, to assume a
relatively trivial picture of steady intra-cavity photon number $|a|^{2}$
evolved under a large damping rate $\kappa$, though there have been many
interesting theoretical studies involving the regime, such as the relaxation
of the requirement $\omega_{m}\gg\kappa$ for realizing quantum effects (see.
e.g. [33, 34, 35, 36, 37, 38, 39, 40]) and the improvement of quantum
efficiencies with multi-mode OMSs (see, e.g. [41, 42, 43, 44]). A nontrivial
scenario of nonlinearity known in unresolved sideband regime is the creation
of pulsed cavity field under stronger driving fields [45, 46, 47]. However, as
we will learn from the later discussion, no resonance phenomenon exists for
any unresolved-sideband OMS driven by an optical field of single frequency.

Our concerned resonance phenomenon emerges when a suitable unresolved-sideband
OMS is driven by two fields (or a two-tone field) with the difference of their
frequencies matching the mechanical frequency of the system. At this resonance
point, the system exhibits a totally different dynamical pattern of boundless
energy harvesting from any other dynamical pattern at a point off the
resonance. Boundless energy harvesting in spite of a nonzero damping rate for
an oscillator was known to be possible by means of parametric resonance [2] or
both classical [48, 49] and quantum version [50, 51] of autoresonance (see
[52] for an overview) under a chirped driving force (its frequency is a
function of time) which brings the oscillator to higher energy constantly. We
realize a boundless energy harvesting through a different mechanism. It is
found that the difference of two drive tones works as a control parameter that
can trigger a mechanism of organizing the motions of two subsystems (the
cavity field and mechanical oscillator) of an OMS. Once this parameter is
tuned to be close to the mechanical frequency of an unresolved-sideband OMS,
the system will enter a pattern of cooperative energy transfer between its two
subsystems, so that this classical dynamical system exhibits a repeatedly
"quantized" energy addition of almost fixed amount in each mechanical
oscillation cycle. The realized step-like energy increment to the mechanical
oscillator is even more regular than those in a corresponding process of
boundless energy harvesting through a quantum autoresonance [50, 51]. This
previously unknown mechanism has a capability of significantly lowering the
power to generate the optical pulse trains applied in communications and
precise measurements.

The rest of the paper is organized as follows. After an illustration of the
dynamical model based on the realistic systems in Sec. 2, we present in Sec. 3
the detailed discussion on the features of the concerned phenomenon and the
existence of a mechanism behind the phenomenon. In the following section, Sec.
4, we describe the phenomena when the system parameters are modified to lose
the mentioned resonance. Some other issues, such as an interesting feature of
preserving the resonance under the parallel shifts of two drive tones, the
perturbation from thermal noise, and the effects of the phase mismatch between
two drives, are discussed in Sec. 5. By Sec. 6 and Sec. 7, respectively, we
provide a feasible application of the resonance phenomenon and provide
necessary information about the experimental implementation of this dynamical
scenario, before we conclude the work with the last section.

Figure 1: Model and phenomenon. (a) An exemplary setup driven by the fields
under a frequency condition. (b) The model of two oscillators respectively
trapped in the potential well $\frac{1}{2}\hbar\omega_{c(m)}X^{2}_{c(m)}$ and
coupled through the potential in Eq. (1). When two driving fields are tuned to
a resonance condition, the repeated excitation of the first one around each
moment $nT$ becomes highly coordinated with the motion of the second, which is
like being constantly excited to the equally distanced energy levels after
each period $T$, though the system dynamics is classical by nature. (c) The
nonlinear resonance. All simulated processes are from the initial condition
$X_{m}(P_{m})=0$, $|a|^{2}=0$, and are obtained with $g_{m}/\kappa=10^{-5}$,
$\omega_{m}/\gamma_{m}=10^{6}$, $E/\kappa=2\times 10^{5}$,
$\omega_{1}=\omega_{c}$, and $\omega_{2}=\omega_{c}-\omega_{m}$. No specific
$\omega_{c}$ is necessary by working with Eq. (3). (d) The stair slope
tendency with the ratio $g_{m}E/\kappa^{2}$ ($g_{m}$ is fixed), for the
mechanical frequencies used in (c).

##  2 Model

We illustrate with a Fabry-Perot-type OMS in Fig. 1(a). Under the radiation
pressure from the cavity photons with the number $|a|^{2}$, the energy
$\hbar\omega_{c}|a|^{2}$ stored between two mirrors, which form a cavity with
its resonance frequency $\omega_{c}$, will be modified to
$\hbar\omega_{c}(1-x_{m}/L)|a|^{2}$ after a tiny displacement
$x_{m}(t)=\sqrt{\hbar/(m\omega_{m})}X_{m}(t)$ ($X_{m}(t)$ is the dimensionless
displacement) of the end mirror with an effective mass $m$. This modification
is valid under the condition $|x_{m}|\ll L$ (the size of the cavity), and it
gives rise to an interaction potential

$\displaystyle V_{int}(t)=-\hbar g_{m}X_{m}(t)|a(t)|^{2}$ (1)

between the cavity field and mechanical mode, where
$g_{m}=\sqrt{\frac{\hbar}{m\omega_{m}}}/L\times\omega_{c}$ is the
optomechanical coupling constant. Then, by modeling the cavity field as
another oscillator with its two perpendicular quadratures being the
dimensionless displacement and momentum ($X_{c}$ and $P_{c}$), such that
$a=(X_{c}+iP_{c})/\sqrt{2}$, one has the nonlinear dynamical equations

$\displaystyle\dot{a}=-\kappa
a-i(\omega_{c}-g_{m}X_{m})a+E(e^{-i\omega_{1}t+i\theta_{1}}+e^{-i\omega_{2}t+i\theta_{2}})$
$\displaystyle\dot{X}_{m}=\omega_{m}P_{m},$
$\displaystyle\dot{P}_{m}=-\gamma_{m}P_{m}-\omega_{m}X_{m}+g_{m}|a|^{2}+\sqrt{2\gamma_{m}}\xi_{m}(t)$
(2)

for the model in Fig. 1(b), where $\gamma_{m}$ is the mechanical damping rate,
$\omega_{1(2)}$ the drive frequency, and the used dimensionless mechanical
momentum $P_{m}(t)$ is related to the real momentum $p_{m}(t)$ as
$p_{m}(t)=\sqrt{\hbar m\omega_{m}}P_{m}(t)$. The cavity field damping rate
$\kappa=\kappa_{e}+\kappa_{i}$ includes two parts; $\kappa_{e}$ measures the
coupling to the driving field and $\kappa_{i}$ indicates the intrinsic loss.
In the regime where $\omega_{m}>\kappa$, the pump power
$\hbar\omega_{1(2)}E^{2}/(2\kappa_{e})$ should be higher than a threshold of
Hopf bifurcation to drive an OMS to oscillate. Before Sec. 5 we neglect the
thermal noise term $\sqrt{2\gamma_{m}}\xi_{m}(t)$ and assume an identical
driving field phase $\theta_{1}=\theta_{2}=0$.

By a transformation $a\rightarrow ae^{-i\omega_{c}t}$ of the cavity field
mode, the first equation in Eq. (2) will take the form

$\displaystyle\dot{a}=-\kappa
a+ig_{m}X_{m}a+E\left(e^{i\Delta_{1}t}+e^{i\Delta_{2}t}\right)$ (3)

in the reference frame rotating at the cavity frequency $\omega_{c}$, where
$\Delta_{1(2)}=\omega_{c}-\omega_{1(2)}$ is the detuning of one drive tone
from the resonant frequency $\omega_{c}$. It will not modify the other
equations in Eq. (2). Then, no specific resonant cavity $\omega_{c}$ will be
needed in the numerical integral of the dynamical equations, while the damping
rate $\kappa$ is used to scale all other parameters in such equations
reformatted with a dimensionless evolution time $\kappa t$.

In the numerical calculations we adopt the mechanical quality factor
$Q=\omega_{m}/\gamma_{m}$ in the range $10^{3}-10^{6}$, the optomechanical
coupling constant $g_{m}=10^{-5}\kappa$, and the mechanical frequency in the
order of $\omega_{m}\sim 0.1\kappa$. If the optical damping rate $\kappa$ is
in the order of $2\pi\times 10$ MHz, the required coupling $g_{m}$ is only in
the order of $2\pi\times 100$ Hz. These parameters are well within those
achieved in the past experiments reviewed in Ref. [7]. Because our concerned
systems are of unresolved sideband, the requirements on their fabrication and
performance are much less demanding. More details in this aspect are given in
Sec. 6 and Sec. 7.

Figure 2: Comparison between two scenarios. (a) The simultaneously evolving
mechanical energy, cavity photon number, and mechanical momentum. Under the
continuous pumping there is only an averaged $|a|^{2}\sim 10^{4}$ (invisible
by the used scale) between two pulse peaks. The used OMS has
$\omega_{m}/\kappa=0.2$, $\gamma_{m}/\omega_{m}=10^{-6}$, and
$g_{m}/\kappa=10^{-5}$, while the pumps are with $E/\kappa=2\times 10^{5}$ and
$\omega_{1}=\omega_{c}$, $\omega_{2}=\omega_{c}-0.25\kappa$. (b) The
corresponding evolution processes due to a difference
$\omega_{2}=\omega_{c}-0.2\kappa$. Here, the mechanical momentum $P_{m}$
evolves smoothly and an obvious change of its amplitude should be seen over a
longer evolution duration $\kappa\Delta t$. Figure 3: Generated pulses from
drive-frequency match. (a) The pulses of the exemplary systems
[$\omega_{m}/\kappa=0.1$ (red), $0.2$ (blue) and $0.5$ (black)], which drop at
the same pace after each period $T$. The corresponding step heights vary
within an order of $1\%$ at the beginning of a dynamical process, so that
those in Fig. 1(c) (except for the first steps) look rather uniform. The
evolution time is counted by the oscillation period number
$t/T\sim\omega_{m}t$. (b) The frequency comb of the IPP pulse due to
$\omega_{m}/\kappa=0.1$. We here fix $g_{m}/\kappa=10^{-5}$ and
$\gamma_{m}/\kappa=10^{-5}$, and use two drives with $E/\kappa=10^{5}$ and
$\omega_{1}=\omega_{c}$, $\omega_{2}=\omega_{c}-\omega_{m}$.

##  3 Dynamical pattern in resonance

###  3.1 Relevant phenomena

A particular choice for the setup in Fig. 1(a) is that the two driving fields
with the same amplitude $E$ keep their frequency difference as
$|\omega_{1}-\omega_{2}|=\omega_{m}$. Under this condition, a resonance
phenomenon manifests to have the mechanical energy
$\mathcal{E}_{m}(t)=1/2\left(X_{m}^{2}(t)+P_{m}^{2}(t)\right)$ evolving as the
temporal stairs in Fig.1(c). While the length of their steps is the same as
the corresponding mechanical oscillation period $T=2\pi/\omega_{m}$, a
temporal stair for each fixed $\omega_{m}$ becomes steeper with the nonlinear
magnitude $g_{m}$ enhanced by the drive amplitude $E$ [see Fig. 1(d)]. The
energy transfer is like a repeated quanta
$\hbar\omega_{m}\delta\mathcal{E}_{m}$ ($\delta\mathcal{E}_{m}$ is the step
height equivalent to a phonon number) from the cavity field to the mechanical
oscillator in each period $T$. Viewed as a separate part, the mechanical
motion takes the form

$\displaystyle P_{m}(t)$ $\displaystyle=$ $\displaystyle
g_{m}\int_{0}^{t}d\tau
e^{-\gamma_{m}(t-\tau)}\\{\cos\frac{1}{2}\sqrt{4\omega_{m}^{2}-\gamma_{m}^{2}}(t-\tau)$
(4) $\displaystyle-$
$\displaystyle\frac{\gamma_{m}\sin\sqrt{4\omega_{m}^{2}-\gamma_{m}^{2}}(t-\tau)}{\sqrt{(4\omega_{m}^{2}-\gamma_{m}^{2})}}\\}|a(\tau)|^{2}$

from the last two equations in Eq. (2), necessitating a pulsed field intensity
$|a(t)|^{2}$ for the step-like change of the energy $\mathcal{E}_{m}(t)$.
Pulsed cavity field due to pumping unresolved-sideband OMSs
($\omega_{m}<\kappa$) by a single-frequency field was theoretically predicted
[45, 46, 47] and experimentally observed [53]. However, under two drives
satisfying the condition $|\omega_{1}-\omega_{2}|=\omega_{m}$, why there
exists a particular evolution pattern demonstrating the temporal stairs with
their uniform steps in Fig. 1(c) should be better understood from comparing
two scenarios simulated in Fig. 2.

The first scenario, with $|\omega_{1}-\omega_{2}|$ mismatching $\omega_{m}$ by
$\delta=0.05\kappa$ ($|\omega_{1}-\omega_{2}|=\omega_{m}+\delta$), gives the
pulsed cavity field intensities $|a(t)|^{2}$ around each half mechanical
oscillation period $t=nT/2$ ($n$ is an integer) as in Fig. 2(a). These pulses
provide the in-phase pushes (IPP) when they act along the same direction of
the moving oscillator (around each positive maximum $P_{m}$), but work as the
out-of-phase pushes (OPP) when they oppose the mechanical motion around each
negative maximum $P_{m}$. The real-time energy $\mathcal{E}_{m}(t)$ thus
undergoes alternate jumps because the pulse height changes with its period
$(\omega_{m}/\delta)T=4T$ in this situation. There are other variation
patterns of the mechanical energy and the pulses off the resonance point
$|\omega_{1}-\omega_{2}|=\omega_{m}$, as we will see in Sec. 4.1. To the
second resonance scenario ($\delta=0$) in Fig. 2(b), the OPP pulses are highly
suppressed but the IPP pulses seem to stabilize gradually as in Fig. 3(a). The
IPP pulses sustain the repeated energy addition
$\hbar\omega_{m}\delta\mathcal{E}_{m}$ to the oscillator until its
displacement $X_{m}$ goes beyond the valid range for the potential in Eq. (1),
and their Fourier transform in Fig. 3(b) shows an optical frequency comb (OFC)
with its teeth distanced by the mechanical frequency $\omega_{m}$.

Figure 4: Comparison between the total mechanical energy and its kinetic part.
The kinetic energy $\mathcal{E}_{k}(t)=\frac{1}{2}P_{m}^{2}(t)$ reaches its
maximum whenever the mechanical oscillation is at each half cycle $t=nT/2$
($n$ is an integer), along with the evolution of the total mechanical energy
$\mathcal{E}_{m}(t)=\frac{1}{2}\left(X_{m}^{2}(t)+P_{m}^{2}(t)\right)$. In the
middle of each step for the mechanical energy $\mathcal{E}_{m}(t)$, the action
of the pulsed field is in the opposite direction of the mechanical motion,
while the pulsed field speeds up the mechanical oscillator around each step
jump of the mechanical energy. This comparison clearly shows that the
dimensionless momentum $P_{m}(t)$ around each step jump of the energy
$\mathcal{E}_{m}(t)$ is higher than the $P_{m}(t)$ around a previous one. Here
the system parameters are the same as those in Fig. 2(b). Figure 5: A self-
organized cooperation between two subsystems. (a) The fitting of the
mechanical energy $\mathcal{E}_{m}(t)$ has a completely linear tendency over a
sample duration $\kappa\Delta t=3000$ from the moment $\kappa t=10000$ to
another moment $\kappa t=13000$. (b) and (c) The details of three steps,
around $\kappa t=10100$ (red solid), $\kappa t=10100+100\pi$ (blue dot-
dashed), and $\kappa t=10100+200\pi$ (black dashed), extracted along the
fitted straight line in part (a), together with the associated cavity field
pulses that create these steps. The steps and their corresponding pulsed field
intensities $|a|^{2}$ are all displaced to the common starting moment of a
reference time $t_{r}$ for the purpose of their comparison. The insets show
the details in the developments of the temporal steps and the corresponding
field intensities $|a|^{2}$. Each stabilized part of the steps [beyond Fig.
5(b)] completely coincides to realize the same step height and the linear
tendency in (a). The exemplary OMS under the drives with the same amplitude
$E/\kappa=10^{5}$ has the same systems parameters as those in Fig. 2.

###  3.2 Example of field-oscillator cooperation

In terms of the model in Fig. 1(b), the second scenario in Fig. 2 is that the
oscillator modeling the cavity field is excited around each moment $t=nT$ and,
after the duration of the pulsed $|a(t)|^{2}$, it drops back to near the
bottom of the binding potential. The excited one happens to bring an impulse
$I_{p}$ to the other one around its highest positive speed at $X_{m}(nT)=0$,
so this mechanical oscillator will go faster whenever it cycles back to the
point $X_{m}((n+1)T)=0$ in the next oscillation period (for an oscillator with
high quality factor the energy loss in each oscillation period is much less
than the energy added by the pulsed field). This fact can be seen more clearly
from Fig. 4 showing the comparison between the dimensionless kinetic energy
$\mathcal{E}_{k}(t)=1/2P_{m}^{2}(t)$ of the oscillator and its whole
mechanical energy
$\mathcal{E}_{m}(t)=1/2\left(X_{m}^{2}(t)+P_{m}^{2}(t)\right)$. Moreover, the
momentum $P_{m}(t)$ of the mechanical oscillator keeps a smooth evolution
during its collision with the pulsed field; compare the evolution course of
$P_{m}(t)$ with that of $\mathcal{E}_{m}(t)$ in Fig. 2(b).

The realized interaction under the condition
$|\Delta_{1}-\Delta_{2}|=\omega_{m}$ for two driving fields brings about a
perfect linear tendency for the evolution of the mechanical energy
$\mathcal{E}_{m}(t)$; the completely straight line [the dashed one shown in
the inset of Fig. 5(a)] fitted for the energy $\mathcal{E}_{m}(t)$ lasts over
a duration $\kappa\Delta t=3000$ for the example in Fig. 5(a). It means that
the mechanical energy $\mathcal{E}_{m}(t)$ keeps jumping along the straight
stair by each constant step height $\delta\mathcal{E}_{m}$ around the moments
$nT$ ($n$ is an integer). Suppose that the $n$-th tall pulse in Fig. 2(b) acts
within a small time window $t\in[nT-\delta t_{n,1},nT+\delta t_{n,2}]$ near
the moment $nT$ (the $n$-th oscillation cycle from the beginning of the time
evolution), to have its whole duration $\delta t_{n}=\delta t_{n,1}+\delta
t_{n,2}$. Multiplying $dX_{m}(t)$ on both side of the third equation in Eq.
(2) and performing an integral over the pulse duration $\delta t_{n}$, one
will have

$\displaystyle\delta\mathcal{E}_{m}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(P^{2}_{m}(nT+\delta
t_{n,2})+X^{2}_{m}(nT+\delta t_{n,2})\right)$ (5) $\displaystyle-$
$\displaystyle\frac{1}{2}\left(P^{2}_{m}(nT-\delta
t_{n,1})+X^{2}_{m}(nT-\delta t_{n,1})\right)$ $\displaystyle=$ $\displaystyle
g_{m}\int_{nT-\delta t_{n,2}}^{nT+\delta t_{n,1}}dt|a(t)|^{2}P_{m}(t)$

as the step height of the mechanical energy $\mathcal{E}_{m}(t)$ around the
moment $t=nT$. Considering the fact $\gamma_{m}\ll\omega_{m}$, we here neglect
the contribution from the mechanical damping force during the small time
window $\delta t_{n}$. The momentum $P_{m}(t)$ after one more oscillation
period (the $P_{m}(t)$ for $t\in[(n+1)T-\delta t_{n+1,1},(n+1)T+\delta
t_{n+1,2}]$) is higher than the $P_{m}(t)$ for $t\in[nT-\delta
t_{n,1},nT+\delta t_{n,2}]$, as seen from Fig. 4. To have the same definite
integral around these two moments $t=nT$ and $t=(n+1)T$ according to Eq. (5)
so that the mechanical energy jumps around the two moments have the same
height, the pulsed field intensity $|a(t)|^{2}$ should adjust itself following
the increased momentum $P_{m}(t)$ around the next step jump of the energy
$\mathcal{E}_{m}(t)$. Then it will build up a kind of their synchronization
within the range like the one in Fig. 5(a). The effect of such synchronization
is that the pulsed field lasting for each short period $\delta t_{n}$ ($n$ as
a series of integers) happens to do the exactly same amount of work
($\delta\mathcal{E}_{m}=\text{constant}$) to the mechanical oscillator, which
runs faster and faster.

Figure 6: Resonance and its breakdown. (a) A comparison between the single-
drive scenarios and the resonance under two simultaneous drives satisfying
$|\omega_{1}-\omega_{2}|=\omega_{m}$. The nearly identical
$\mathcal{E}_{m}(t)$ for both individual drives with their different $\Delta$
is bound as the illustrated asymptotic stability. We adopt the definition
$\Delta=\omega_{c}-\omega$, and the system has $\omega_{m}/\kappa=0.2$,
$g_{m}/\kappa=10^{-5}$, and $\gamma_{m}/\kappa=10^{-5}$. (b) The corresponding
evolution courses of the energy $\mathcal{E}_{m}(t)$ after
$\Delta_{1}\rightarrow\Delta_{1}$, $\Delta_{2}\rightarrow\Delta_{2}+\delta$,
with $\delta=-10^{-4}\kappa$. The lower quality factors
$Q=\omega_{m}/\gamma_{m}$ make the courses deviate from the linear tendency
more obviously. (c) The allowed maximum error $\delta_{c}$ to have the
temporal stairs extending over $\kappa t=1.5\times 10^{4}$. Here
$E/\kappa=10^{4}$. (d) The maximum duration $\kappa t_{c}$ of the temporal
stairs after an existing error $\delta$ from the drive-frequency condition.
Decreasing or increasing $g_{m}$ does not change the tendency so much.

To see how a pulsed field force precisely add a constant amount of energy or
do the same amount of work during each small time window $\delta t_{n}$ within
the illustrated range, we pick out three equally distanced steps along the
fitted straight line on the temporal stair in Fig. 5(a), and displace these
steps and the associated pulses that create them to a common staring point to
compare their respective dynamical evolution process in Figs. 5(b) and 5(c).
From one sample process to another, one finds that the pulses shown in Fig.
5(c) are fine-tuning their fronts by themselves so that the rise of the peak
for a pulse that pushes the faster oscillator slightly delays by a proper
amount; the details are illustrated in the inset of Fig. 5(c). Due to such a
delicate adjustment, the different energy additions to the mechanical
oscillator, as the heights of the temporal steps, happen to be the same—the
three evolved horizontal lines in the upper right inset of Fig. 5(b)
completely overlap. For the example in Fig. 2(b), the step height
$\delta\mathcal{E}_{m}=5.623\times 10^{11}$ near $\kappa t=10^{3}$ only lowers
to $\delta\mathcal{E}_{m}=5.183\times 10^{11}$ till $\kappa t=10^{5}$, as the
self-adjustment of the pulse action also modifies over a larger time scale.
Confined to the straight-line increase of mechanical energy
$\mathcal{E}_{m}(t)$ in Fig. 5(a), the coupled pulses and mechanical
oscillator reach a perfect mutual coordination or are locked together into a
unique pattern of energy transfer. This could be viewed as a generalization of
the phase synchronization [54, 55, 56] between two harmonic oscillations—two
totally different types of motion (one is harmonic oscillation but the other
is pulsed excitation) are self-organized into a unique way of energy transfer.
In nonlinear dynamical processes it is common to see the locking phenomena in
a parameter space (the well-known Devil’s staircase) or a locked step-by-step
evolution of oscillation phase (see, e.g. [54]), but a perfect cooperation
between two subsystems to realize a constantly and uniformly step-like energy
addition to one of them is beyond the previously studied phenomena of
synchronization [54, 56].

###  3.3 Pattern locking mechanism

Driven by a single pump, the responding mechanical motion never reaches a
resonance. Unlike in the regime $\omega_{m}/\kappa>1$, the induced mechanical
motion in the regime $\omega_{m}/\kappa<1$ can be nearly identical when
scanning the single-drive frequency from a blue-detuned to a red-detuned; see
one example in Fig. 6(a). In contrast, two simultaneous drives lead to a
boundless energy harvesting at the point $|\omega_{1}-\omega_{2}|=\omega_{m}$,
due to a triggered mechanism to lock the system into a cooperative energy
transfer. This mechanism even works under the drives of very low power
$E/\kappa\ll 1$ (see Sec. 4.2), though the associated cavity fields are not
pulses. A Hopf bifurcation from a steady to an oscillating OMS occurs when the
amplitude $E$ of a single drive should be over a threshold for the OMSs in the
regime $\omega_{m}/\kappa>1$, but the above-mentioned mechanism brings an
unresolved-sideband OMS to oscillation without a restriction on drive power.
The dynamical pattern of resonance cannot be preserved and gradually turns
into something like the one in Fig. 2(a) if $|\omega_{1}-\omega_{2}|$ keeps
going away from $\omega_{m}$, but it can be compensated to some extent by a
higher mechanical quality factor, as shown from Figs. 6(b) and 6(c). In the
ideal situation $\gamma_{m}=0$, the perfect stairs of $\mathcal{E}_{m}(t)$
last longer than $\kappa t=10^{4}$ in spite of an error
$|\delta|=10^{-4}\kappa$ from the condition
$|\omega_{1}-\omega_{2}|=\omega_{m}$, under which the linearly rising tendency
of $\mathcal{E}_{m}(t)$ also disappears due to a larger damping rate
$\gamma_{m}$ (see Sec. 4.3). Fig. 6(d) shows that changing the magnitude
$g_{m}$ itself cannot improve the process. All these facts further prove that
the mechanism of pattern locking is rather different from all other types of
nonlinearity; a mismatch between $|\omega_{1}-\omega_{2}|$ and $\omega_{m}$ or
more energy loss will finally impair the cooperation between two subsystems.

##  4 Scenarios beyond perfect resonance

###  4.1 Dynamical patterns off resonance point

In practice it is much easier to have two pumping tones off the resonant
point, i.e. $|\omega_{1}-\omega_{2}|=\omega_{m}+\delta$ with an error
$\delta$. If the error is tiny, such as $|\delta|=10^{-4}\kappa-10^{-3}\kappa$
for the examples in Fig. 6, the mechanical energy evolution will become a
ladder that gradually bends with time. At somewhere further away from the
resonance point, there will appear rich varieties of local patterns for the
energy $\mathcal{E}_{m}(t)$ and its associated cavity field pulses, in
addition to the overall asymptotic stability for the real-time
$\mathcal{E}_{m}(t)$ (the system stabilizes at $t\rightarrow\infty$). In Fig.
7 we only display some patterns for the jumps of the energy
$\mathcal{E}_{m}(t)$, since the associated change of the field pulse height
can be deduced with their one-to-one correspondence as in Fig. 2.

Figure 7: Periodic mechanical energy patterns off the resonance point, The
stabilized period is determined by the ratio $\omega_{m}/|\delta|$. The
evolution in (d) with an irrational ratio $\omega_{m}/\delta$ first undergoes
an aperiodic stage before its stabilization to a periodic pattern. The setup
in these processes has the same parameters as those in Fig. 2 and
$E/\kappa=2\times 10^{5}$.  Figure 8: Scenarios under arbitrary drive power.
(a) The pattern of similar oscillation for the systems under low drive powers
but still meeting the resonance condition. The system parameters for the
exemplary OMS are the same as those in Fig. 2. (b) The comparison between the
effects of drive power for a single tone and a double-tone drive under the
frequency condition for realizing the mentioned resonance. Three patterns
exist in the former, while the latter has only two patterns that gradually
take transition to each other by tuning the drive amplitude $E$.

We find that such dynamical patterns off the resonance point are completely
determined by the ratio $\omega_{m}/|\delta|$. Given a rational ratio
$\omega_{m}/\delta=m/n$, where $m$ and $n$ are two relatively prime integers,
the period for a round of variation of the pulse height, which displays the
patterns illustrated in Fig. 7, is $N(\omega_{m}/\delta)T$ with $N$ being an
integer and $T=2\pi/\omega_{m}$ being the mechanical oscillation period. For
example, the variation period will be $T$, the pattern of the single drives in
Fig. 6(a), if the error $\delta$ is up to $-\omega_{m}$. The exemplary
patterns in Figs. 7(a), 7(b) and 7(c) have their periods
$9(\omega_{m}/\delta)T$, $2(\omega_{m}/\delta)T$ and $(\omega_{m}/\delta)T$,
respectively. More interesting is an irrational ratio $\omega_{m}/\delta$ as
in Fig. 7(d). In this situation the system first undergoes an aperiodically
transient stage and then stabilizes to a particular periodic pattern, but the
stabilized period does not follow a fixed law. Because the rational numbers
are like isolated islands in the ocean of the irrational numbers, the period
for the dynamical patterns changes randomly along the axis $\omega_{m}/\delta$
in Fig. 7. Different errors $\delta$ make the system be locked into different
patterns of periodic variation for the pulse height and the mechanical energy
jumps. The resonance point at $\delta=0$ can be regarded to have an infinitely
long period with $\omega_{m}/\delta\rightarrow\infty$.

###  4.2 Mechanical resonance under extremely low drive power

The resonance phenomenon under the condition
$|\Delta_{1}-\Delta_{2}|=\omega_{m}$ exists even when the drive powers are
very low, as shown in Fig. 8(a). In fact, this illustrated scenario of the
similar responses of the system to the varied drive amplitude $E$ even exists
when $E\rightarrow 0$, although the increase of the mechanical energy
$\mathcal{E}_{m}(t)$ loses the linear tendency even at the beginning stage. A
remnant cooperation between two subsystems, due to the action of a mechanism
to lock them into the pattern, still helps to push the mechanical oscillation
to higher amplitude until after a relatively long duration.

On the other hand, when the same system is driven by a single-tone field,
there exists a gradual transition from a quasi-steady (like the observation
reported in Ref. [29]) to an oscillating field once the drive amplitude is
over the level $E/\kappa\sim 10^{3}$ for the used OMS. If the pump power of
the single drive is increased further, the systems in the regime
$\omega_{m}/\kappa<1$ will respond almost identically to the differently
detuned driving fields [see Fig. 6(a)]. In Fig. 8 (b) we summarize the
dynamical patterns due to a varied drive amplitude $E$ in both scenarios of a
single-tone and a double-tone drive meeting the condition
$|\Delta_{1}-\Delta_{2}|=\omega_{m}$. To both of the scenarios, there is no
bifurcation on the way of tuning the drive power because, unlike near a Hopf
bifurcation in the regime $\omega_{m}/\kappa>1$, the gradual transitions
between different patterns do not show a phenomenon of critical slowing-down.

Figure 9: Factors that damage straight stair-like tendency. (a) The evolution
processes for the different $\omega_{m}$, with a vanishing mechanical damping
$\gamma_{m}=0$ for each $\omega_{m}$. (b) The evolution processes for the
different $\gamma_{m}$, but with a fixed mechanical frequency
$\omega_{m}=0.2\kappa$. Here, we consider an OMS with $g_{m}=10^{-5}\kappa$
and apply two driving fields with the same amplitude $E=10^{6}\kappa$ and
their detunings $\Delta_{1}=0$ and $\Delta_{2}=\omega_{m}$.

###  4.3 Stronger spring force and more energy loss

Changing two other parameters can destroy the linearly rising stairs for the
energy $\mathcal{E}_{m}(t)$, even if the system is still under the resonance
condition. In Fig. 9(a) we display a group of sample evolution processes with
their only difference in the mechanical frequency $\omega_{m}$. Here, we
assume an ideal situation $\gamma_{m}=0$ (the dissipation in the system is
only through the cavity field damping $\kappa\neq 0$), so the dynamics is
simplified to see the essential effect due to the mechanical frequency change.
From the numerical simulations of the pulsed $|a|^{2}$ shown in the inset of
Fig. 3(a), one sees that, in the regime $\omega_{m}<\kappa$, the peak heights
of the IPP pulses go down with the mechanical frequency $\omega_{m}$ but those
of the OPP pluses grows up with $\omega_{m}$. Then the evolving
$\mathcal{E}_{m}(t)$ for a higher $\omega_{m}$ will become square-shaped [see
the inset in Fig. 8(a)]. Given a higher $\omega_{m}$, the action of the IPP
pulses also becomes less matched to the mechanical oscillation around its
positive maximum speed, thus losing the dynamical pattern of the cooperation
between two subsystems. Another factor that is illustrated in Fig. 9(b) is the
mechanical damping rate $\gamma_{m}$, which causes the loss

$\displaystyle\Delta\mathcal{E}_{m}=-\gamma_{m}\int_{nT}^{(n+1)T}P_{m}^{2}(t)dt$
(6)

of the mechanical energy
$\mathcal{E}_{m}(t)=1/2\left(X_{m}^{2}(t)+P_{m}^{2}(t)\right)$ in each
oscillation period $T$. At a larger mechanical oscillation amplitude, more
energy loss will be experienced by the oscillator that is running faster to
have higher $P^{2}_{m}$. The input energy from the pulsed field will not be
sufficient to keep the linear increase of the energy $\mathcal{E}_{m}(t)$.

Figure 10: Resonance under parallel drive-frequency shifts. (a) The examples
of the created mechanical energy $\mathcal{E}_{m}$ by the three combinations
$(\Delta_{1},\Delta_{2})=(-0.1\kappa,0)$ (red),
$(\Delta_{1},\Delta_{2})=(-0.05\kappa,0.05\kappa)$ (blue), and
$(\Delta_{1},\Delta_{2})=(0,0.1\kappa)$ (black), which all have the difference
$|\Delta_{1}-\Delta_{2}|$ matching the used mechanical frequency
$\omega_{m}=0.1\kappa$. By employing the single drives, $\Delta=-\omega_{m}$
(pink), $\Delta=0$ (green), and $\Delta=\omega_{m}$ (dark), respectively, the
realized mechanical energy will be those shown in the inset. Here, the drive
amplitude is $E=10^{4}\kappa$. (b) The corresponding result by increasing the
drive amplitude to $E=10^{5}\kappa$. The system has
$Q=\omega_{m}/\gamma_{m}=10^{4}$ and $g_{m}/\kappa=10^{-5}$.

##  5 Discussions on other related issues

###  5.1 Parallel shift of drive frequencies

Previously, OMSs driven by multiple driving fields were studied theoretically
(see, e.g. [57]) and experimentally [58] in the regime where
$\omega_{m}/\kappa>1$. A more interesting scenario that is predicted for the
regime is locking the mechanical oscillation amplitude to some fixed values
under the condition $|\omega_{1}-\omega_{2}|=\omega_{m}$, in spite of varying
the drive power; this scenario realizes a series of fixed stable mechanical
orbits like energy levels [59, 60, 61, 62]. These frozen mechanical orbits
still exist after shifting the frequencies of two driving fields together,
given a possible adjustment of the drive power.

Our concerned resonance phenomenon in the regime of $\omega_{m}/\kappa<1$ is
also preserved under the parallel shift

$\displaystyle\omega_{1}\rightarrow\omega_{1}+\delta,$
$\displaystyle\omega_{2}\rightarrow\omega_{2}+\delta$ (7)

of the frequencies of two driving fields. Once the absolute difference
$|\omega_{1}-\omega_{2}|$ is close to the mechanical frequency $\omega_{m}$, a
suitable system pumped by two drives that are not too weak will demonstrate
the resonance phenomenon. In Fig. 10 we compare the temporal steps of the
energy $\mathcal{E}_{m}(t)$ that are realized by the different combinations of
$\Delta_{1}$ and $\Delta_{2}$ under the condition
$|\Delta_{1}-\Delta_{2}|=\omega_{m}$. These temporal stairs under such
condition are parallel to one another, but the most blued-shifted combination
leads to the highest mechanical energy. To maintain a fixed distance
$|\omega_{1}-\omega_{2}|$ is, therefore, a technical prerequisite to realize
the dynamical scenario. A similar condition was required in a previous
experiment performed in a resolved sideband regime ($\omega_{m}/\kappa>10$)
[58], where two drive tones are tuned to near $\omega_{c}\pm\omega_{m}$ to see
a type of dynamical instability.

As a comparison, an individual action of the involved single driving fields
can give rise to different responses of the mechanical oscillator. The
examples are given in the inset of Fig. 10(a). By increasing the pump power,
the evolution trajectories of $\mathcal{E}_{m}(t)$ will become more and more
identical, as seen from those in the inset of Fig. 10(b) to those in an inset
of Fig. 6(a). The contrast between the doubly driven and the singly driven
scenario in Fig. 10(a) is dramatic. Under that used lower pump amplitude
$E=10^{4}\kappa$ for the exemplary system, any single-tone drive can never
create a pulsed cavity field that realizes the square-shaped
$\mathcal{E}_{m}(t)$ shown in the inset of Fig. 10(b). However, a combination
of two tones simply satisfying the condition
$|\Delta_{1}-\Delta_{2}|=\omega_{m}$ will generate a pulsed field that drives
the mechanical oscillator to repeated sudden jumps of its energy. This is
another illustration of enhancing the nonlinearity under the drive-frequency
condition $|\Delta_{1}-\Delta_{2}|=\omega_{m}$.

Figure 11: Insignificant influence of thermal noise. Compared with the results
at zero temperature, the thermal perturbation at the room temperature does not
bring about much influence on the temporal stairs of the mechanical energy.
The fixed system parameters for the exemplary systems are the same as those in
Fig. 2, and the specific cavity damping rate is $\kappa=2\pi\times 30$ MHz.

###  5.2 Thermal noise effect

The dynamical scenarios related to our concerned resonance are characterized
by gradual transitions, instead of sudden transitions of bifurcation in many
other nonlinear systems. For example, the loss of the linearly increasing
tendency of $\mathcal{E}_{m}(t)$, as the bent temporal stairs displayed in
Fig. 9, is gradual. One consequence of this feature is in the numerical
integral of Eq. (2); all of our numerical calculations are not obviously
affected by the adopted precision, in contrast to those near a bifurcation
point where the critical dynamics makes the simulation results highly
sensitive to the used computation precision (see, e.g. Supplemental Material
of [15]).

Noise perturbations are also less important to the concerned dynamical
process. To study a specific example, we adopt a thermal noise term in Eq.
(2). The thermal noise at room temperature can be approximated by a white
noise with the correlation $\langle\xi_{m}(t)\xi_{m}(t^{\prime})\rangle\approx
2n_{th}\delta(t-t^{\prime})$ ($n_{th}$ is the thermal occupation at a certain
temperature) [63]. This noise term can be simulated by a random function [15].
In Fig. 11 we display the evolving stairs of the mechanical energy
$\mathcal{E}_{m}(t)$ at the room temperature $300$ K for two exemplary
$\omega_{m}$. One sees that the influence of the environmental temperature is
negligible. If we reduce the mechanical quality factor
$Q=\omega_{m}/\gamma_{m}$ by $10^{3}$ times, the realized step heights will be
slightly lowered. The dynamical pattern with the cooperative energy transfer
between cavity field excitation and mechanical oscillation is rather robust
against thermal perturbation.

Figure 12: Effect of unmatched driving field phases. (a) A small deviation
$\phi=0.01$ can hardly impair the dynamical pattern that realizes the
repeatedly uniform steps of the mechanical energy. (b) and (c) The nonlinear
interference results due to the large relative phases $\phi$. Different
patterns of the pulsed field, corresponding to the illustrated mechanical
energy, can be created by adjusting the phase. The system parameters for the
exemplary OMS are the same as those in Fig. 2.

###  5.3 Phase mismatch between driving fields

The two fields as the driving terms in Eq. (2) may have different phases
$\theta_{1}$ and $\theta_{2}$ in reality. We here illustrate the effect of the
relative phase $\phi=|\theta_{1}-\theta_{2}|$ by the numerically calculated
$\mathcal{E}_{m}(t)$ and $|a(t)|^{2}$ in Fig. 12. Deviating the phase $\phi$
from the ideal condition $\phi=0$, one will obtain various results. For
example, in Fig. 12(a), the phase mismatch in the order of $10^{-2}$ does not
bring an obvious difference to the coordinated pattern demonstrating the
mechanical energy steps and their corresponding pulse train. Compared to the
drive-frequency match condition $|\Delta_{1}-\Delta_{2}|=\omega_{m}$, which
demands fitting two driving fields to a quantity of another subsystem (the
frequency $\omega_{m}$ of the mechanical oscillator), phase match for these
two fields is much less rigorous. In experimental implementation of the
dynamical scenario it is therefore much easier to have two driving field
phases unmatched within a considerable range. As the relative phase $\phi$ is
increased further, the resulting dynamical patterns will become rich. Two
typical situations are $\phi=\pi/2$ and $\phi=\pi$, which are shown in Fig.
12(b) and Fig. 12(c), respectively. The OPP pulses, which act to the opposite
moving direction of the mechanical oscillator, cannot be highly suppressed in
these situations, so that the energy $\mathcal{E}_{m}(t)$ displays the square-
shaped evolution courses.

##  6 Application

In an unresolved sideband regime $\omega_{m}<\kappa$, one will see a pulsed
field after plugging a mechanical oscillation $X_{m}(t)=A\sin\omega_{m}t$ with
a large amplitude $A$ into the first equation of Eq. (2). Such pulsed fields
were discussed with possible applications [46, 47]. Especially in a recent
experiment [53], the OFC with $938$ optical comb lines was generated by
pumping a nearly unresolved sideband system with a single blue-detuned drive.
Based on the full dynamics of only one drive in Eq. (2), one sees in Fig.
13(a) the transition from a field of discrete sidebands to the pulsed ones via
increasing the drive power. To create the pulse train on the second row of
Fig. 13(a), the power of a pump should be over $330$ mW by employing a setup
with $\kappa_{e}=2\pi\times 25$ MHz, $\kappa_{i}=0.04\kappa_{e}$, and its
resonant cavity frequency $\omega_{c}=2\pi\times 190$ THz, which are close to
the system parameters of the experimental setup in Ref. [53]. Simply by adding
another drive satisfying $|\omega_{1}-\omega_{2}|=\omega_{m}$, the power for
generating a pulse train can be reduced by at least $1000$ times as shown in
Fig. 13(b).

In Fig. 13(c), the pump power is reduced to the level such that the system
under a single drive only exhibits a quasi-linear response of approximately
steady field intensity, similar to a previous experiment (that setup has
$\omega_{m}\sim 0.001\kappa$) [29]. However, when we apply two drive tones
matching their difference with the mechanical frequency to the same OMS, a
pulse train will surprisingly emerge with their peaks higher than the
stationary field intensity realized under either of the individual drive tones
alone. The pump power to obtain the pulse train will be about $130$ $\mu$W
given the above-mentioned setup close to the experimental one in Ref. [53].
Equal-distanced and approximately identical pulses are generated; for the
example in Fig. 5(a), the variation of the pulse front is within the order of
$\kappa\delta t\sim 0.1$ after the constantly self-adjusted pulse action over
the range $\kappa\Delta t=3000$. Moreover, the frequency span of the generated
OFC can be widened quickly by increasing the drive power.

Figure 13: Cavity fields created under varied drive power. (a) The samples of
the asymptotically stabilized field intensity $|a(t)|^{2}$, driven by a single
field of resonant frequency $\omega=\omega_{c}$. From the top to the bottom,
$E/\kappa=10^{4}$, $5\times 10^{4}$, and $2\times 10^{5}$. (b) The
corresponding results by two drives under the condition
$|\omega_{1}-\omega_{2}|=\omega_{m}$, but with the drive powers being all
reduced by $1000$ times ($E/\kappa=10^{2.5}$, $5\times 10^{2.5}$, and $2\times
10^{3.5}$, respectively). (c) A significantly enhanced optomechanical
nonlinearity. Any of the single drives with the amplitude $E/\kappa=10^{3}$
only leads to a quasi-linear response, but their joint action under the
condition $|\omega_{1}-\omega_{2}|=\omega_{m}$ creates a pulse train. The used
OMS has $\omega_{m}/\kappa=0.2$, $g_{m}/\kappa=10^{-5}$, and
$\gamma_{m}/\kappa=10^{-5}$.

##  7 Experimental feasibility and possible extensions

Theoretically, any OMS, which has a sufficiently high mechanical quality
factor $Q=\omega_{m}/\gamma_{m}$ together with a mechanical frequency
$\omega_{m}$ lower than the cavity field damping rate $\kappa$ by a certain
amount, can realize the described phenomena. The most essential requirement is
to well keep the difference between two drive tones. In experiment the
resonant cavity frequency will drift from the original $\omega_{c}$ due to
changed temperature in operation [53], thus adding a common detuning $\delta$
to the driving fields. However, the same phenomenon still exists after a
parallel shift in Eq. (7), With the available techniques [64, 65], the
frequency difference between two driving lasers can be kept to a good extent.
Two other factors, the thermal perturbation at room temperature and the phase
mismatch between two driving fields, are not important to the dynamical
process as we have clarified in Sec. 5.2 and Sec. 5.3. Because of an enhanced
nonlinearity, the required ratio
$E/\kappa=\sqrt{2\kappa_{e}P/[(\kappa_{e}+\kappa_{i})^{2}\hbar\omega]}$ for
implementing the concerned scenarios under a fixed power $P$ can be much
lowered, thus dispensing with the necessity for very high cavity finesse
($\kappa_{i}$ can be higher).

Upon entering the resonance, the system operates in a stable pattern but in a
dynamical instability with the mechanical amplitude approximately proportional
to $\sqrt{\kappa nT}$ after $n$ mechanical oscillation periods. In the extreme
situation that the mechanical oscillation amplitude becomes compatible with
$\lambda_{c}/2=\pi c/\omega_{c}$, more cavity field modes neighboring the
original one $a=a_{0}$ with its frequency $\omega_{c}=cN\pi/L$ ($N$ is the
node number of the standing wave in the cavity) will be excited [66], and the
potential in Eq. (1) will be modified to

$\displaystyle
V_{int}=\hbar\left\\{\frac{c(N+n)\pi}{L+x_{m}}-\frac{c(N+n)\pi}{L}\right\\}|a_{n}|^{2}$
$\displaystyle=\hbar\left\\{\frac{\omega_{c}+\frac{cn\pi}{L}}{1+\eta_{m}X_{m}}-(\omega_{c}+\frac{cn\pi}{L})\right\\}|a_{n}|^{2}$
(8)

for each excited cavity field mode $a_{n}$ of the system in Fig. 1(a), where
$n=0,\pm 1,\pm 2,\cdots$, and $\eta_{m}=\sqrt{\frac{\hbar}{m\omega_{m}}}/L$.
This multi-mode extension for a previously studied scenario under a single
drive [45] is elaborated in Ref. [66], which shows that its important features
(including the pulsed cavity field corresponding to the saw-tooth-edged
ellipses in the limit cycles of the mechanical motion) are well preserved
after such extension to multiple cavity field modes. It is expected that the
main features of our concerned dynamical behaviors can be also seen after a
similar generalization. To be comparable with the majority of other works
adopting the parameter $g_{m}$ and avoid introducing a new parameter
$\eta_{m}$ from Eq. (8), we work under the condition $|x_{m}|\ll L$ for Eq.
(2), which can be valid for a sufficiently long duration of time to many
realistic experimental setups different from the one in Fig. 1. For example,
the ratio $g_{m}E/\kappa^{2}$ in Fig. 1(d) can be lowered to $0.1$ when
realizing the dynamical scenario with $\omega_{m}=0.1\kappa$. Given the
parameters $g_{m}/\kappa=10^{-5}$ and $E/\kappa=10^{4}$ for such a setup, one
still has a mechanical oscillation amplitude less than $1$ nm after $\kappa
t=10^{5}$, considering the zero-point fluctuation amplitude
$\sqrt{\hbar/(m\omega_{m})}\sim 0.1$ fm and the size $L\sim 30$ $\mu$m of an
available micro-cavity [53].

Another possible extension to multi-mode OMSs is through adding more
mechanical elements or more optical cavities, similar to some previously
studied systems in, for examples. Refs. [67, 68, 69, 70, 71, 72, 73, 74, 75,
76, 77, 78, 79, 80]. This type of extensions may bring about the
synchronization between the mechanical elements [13, 14, 16] in addition to a
field-oscillator synchronization, as well as other nontrivial phenomena such
as the sudden transitions between dynamical patterns [15]. Further researches
will be undertaken to unveil the richer nonlinear dynamical behaviors that may
exist in multi-mode systems.

##  8 Conclusion

We have presented a detailed study on a special dynamical process in
unresolved-sideband OMSs and its related issues. Both aspects of the
illustrated dynamical scenario are highly meaningful to the current
researches. First, the mechanical energy exhibiting repeatedly uniform jumps
provides an example that even two totally different types of motion can be
synchronized by tuning a control parameter. Such a unique energy transfer
brings something new to the research fields of nonlinear resonance [52] and
synchronization [54, 56]. On the other side, there exists a mechanism to
enhance the nonlinearity under a condition of drive-frequency match, allowing
possible applications at low levels of pump power. A prospect of the wider use
of the OMSs of unresolved sideband, which are less demanding in their
fabrications, may be real in view of their dynamics depicted here.

## Acknowledgements

We thank Dr. Zhen Shen, Dr. Yan-Lei Zhang, Dr. Ming Li, and Dr. Luis Martínez
for the helpful discussions on the relevant experimental issues. This work was
supported by National Natural Science Foundation of China (11574093), Natural
Science Foundation of Fujian Province (2020J01061), ANID Fondecyt Regular
(1221250), and Fondo de Iniciación de Universidad Mayor (PEP I-2019021).

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