††thanks: These authors contribute equally to this work††thanks: These authors
contribute equally to this work

# Chiral Chaos Enhanced Sensing

Yun-Qiu Ge  School of integrated circuits, Tsinghua University, Beijing
100084, China  Frontier Science Center for Quantum Information, Beijing, China
Zhe Wang  Department of Automation, Tsinghua University, Beijing 100084, China
Qian-Chuan Zhao  Department of Automation, Tsinghua University, Beijing
100084, China  Jing Zhang<EMAIL_ADDRESS>School of Automation
Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China  MOE
Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong
University, Xi’an 710049, China  Yu-xi Liu<EMAIL_ADDRESS>School of integrated circuits, Tsinghua University, Beijing 100084, China
Frontier Science Center for Quantum Information, Beijing, China

###### Abstract

Chirality refers to the property that an object and its mirror image cannot
overlap each other by spatial rotation and translation, and can be found in
various research fields. We here propose chiral chaos and construct a chiral
chaotic device via coupled whispering gallery mode resonators, where routes to
chaos exhibit pronounced chirality for two opposite pumping directions. The
mechanism responsible for this phenomenon is that time-reversal symmetry of
the traveling-wave light fields is broken by the Rayleigh scatterers inserted
in resonators. Combining with the Lyapunov exponents, we propose metrics to
measure the symmetry and chirality between different chaotic dynamics. We find
that such a chiral chaotic device can be applied to achieve sensing with high
sensitivity, wide detectable range, and strong robustness to the phase and
orientation randomness of weak signals. Our work presents a promising
candidate for on-chip sensing and may have applications in quantum networks
and chaotic communications.

Introduction.—The concept of chirality has permeated into various research
fields, e.g., soft matter physics Jmaterchem2006 , chemistry IntJBiomedSci ,
biological science Philo2016 , particle physics RMP922020 , and optics
Nature5412017 ; RMP872015 . Recently, considerable efforts have been devoted
to the research of chirality in light-matter inter-actions AM2107325 ;
book2017 , and a series of promising applications have been achieved.
Representative examples include enantio-sensitive controls NatPhoton13866 ,
optofluidic sorting of particles NatCom2014 , chiral negative refractive index
materials Science2004 , and chiral optical devices Science2016 ; PRL114053903
. Chaos describes certain nonlinear dynamics which is sensitive to initial
conditions, and has broad applications in physics, chemistry, and engineering
chaosbook . Thus, the chaos combining with chirality may result in new
features for chaos applications. Moreover, existing chiral parameters
RevModPhys711745  are not suitable for characterizing and quantifying the
degree of chirality in chaos.

The radiation-pressure-based hybrid systems are ideal platforms for studying
nonlinear physics and chaos NatPhoton103992016 ; PRL1140136012015 ;
PRL111053602 , and can be used for high precision measurements of weak signals
Braginsky1 ; Braginsky2 . Advanced micro-/nano-fabrication technologies make
the sizes of chaos-based sensors reach the micron level RMP861391 , which
utilize the abrupt changes of observables in the vicinity of critical points
of bifurcation or chaos NatureMaterials2011 ; Appl98 ; IEEE46440 . Such
sensors possess the advantages of high sensitivity and fast response, but
their effectiveness is severely limited by the type of detected signals
NatPhoton9151 ; Appl98 . For uncontrolled signals exerting on sensors, e.g.,
optical signals with random phases or mechanical displacements with uncertain
directions, even if their amplitudes exceed a critical threshold, abrupt
changes (e.g., order-to-chaos, bifurcation cascade) do not necessarily occur
Nature363411 . Thus, the occurrence of false negative errors during the
sensing process is hard to avoid, which hinders the applications of chaos-
based sensors NatPhoton9151 .

In this Letter, we propose a controllable chiral chaotic device using
whispering gallery mode (WGM) resonators as controlled objects and tips as
controllers. Chiral chaos is characterized by phase diagrams, Lyapunov
exponents, bifurcation diagrams, and other means M.C.Gutzwiller . Underlying
physics behind chirality is the regulation of time-reversal symmetry by the
photonic gauge potentials generated via tips. We also propose appropriate
parameters to quantify chiral chaos. By tuning the chiral order-to-chaos phase
transition routes, we construct a type of sensing window that is immune to
false negative errors. Through analysis and comparison, we demonstrate the
superiority of our sensing proposal in sensing performance compared to
existing chaos-based proposals.

Figure 1: (a) Schematic diagram of the proposed chiral chaotic device, where
an optomechanical resonator A is coupled to an optical resonator B, and the
pumping field is input into resonator A through the tapered fiber from either
port. Two tips acting as Rayleigh scatterers contact the rim of resonator B.
(b) Loops consisting of optical modes and their couplings. (c) The abilities
of tips to regulate $|\xi|$ and $\varphi$ under common experimental conditions
(see Supplemental Material SupMat  for details), where the green and gray
regions demarcated by the black curve represent the achievable and
unachievable parameter ranges, respectively.

Chiral chaotic device.—As schematically illustrated in Fig. 1(a), two
resonators are coupled via the evanescent fields, and each supports two
counterpropagating optical WGMs Science2014 . We define $\hat{a}_{\rm CW}$ and
$\hat{a}_{\rm CCW}$ ($\hat{b}_{\rm CW}$ and $\hat{b}_{\rm CCW}$) as the
annihilation operators for CW and CCW modes in resonator A (B), with a
degenerate resonance frequency $\omega_{\rm A}$ $(\omega_{\rm\;\;\\!\\!\\!B})$
and damping rate $\kappa$ $(\gamma)$, respectively. Resonator A supports a
mechanical mode described by displacement $\hat{q}$ and momentum $\hat{p}$
operators, with the frequency $\Omega$ and decay rate $\Gamma$. Chaotic
dynamics of the device originates from the nonlinear optomechanical
interactions between the optical modes and the mechanical mode in resonator A,
i.e., $G\;\;\\!\\!\\!(\hat{a}^{{\dagger}}_{\rm CW}\hat{a}_{\rm
CW}+\hat{a}^{{\dagger}}_{\rm CCW}\hat{a}_{\rm CCW})\;\;\\!\\!\\!\hat{q}$,
where $G$ represents the single-photon optomechanical coupling strength
RMP861391 .

Couplings between optical modes in two resonators are schematically shown in
Fig. 1(b) and are described by the effective Hamiltonian ($\hbar=1$)

$\displaystyle\hat{H}_{\rm eff}=\,$ $\displaystyle-J\hat{a}_{\rm
CW}\hat{b}^{{\dagger}}_{\rm CCW}-|\xi|e^{i\varphi}\hat{b}_{\rm
CCW}\hat{b}^{{\dagger}}_{\rm CW}$ $\displaystyle-J\hat{b}_{\rm
CW}\hat{a}^{{\dagger}}_{\rm CCW}-\eta\;\;\\!\\!\\!\hat{a}_{\rm
CCW}\hat{a}^{{\dagger}}_{\rm CW}+\rm H.c.,$ (1)

where $\eta$ is the backscattering strength between CW and CCW modes in
resonator A caused by surface defects RMP872015 , $|\xi|e^{i\varphi}\;\\!\\!$
and $|\xi|e^{-i\varphi}\;\\!\\!$ denote the complex scattering strength
between WGMs in resonator B, in which both $|\xi|$ and $\varphi$ can be tuned
through tips PRL112203901 ; SupMat , and $J$ is the coupling strength between
two resonators.

Figure 2: (a),(b) Phase diagrams for different input ports, with (a) and (b)
corresponding to input from port 1 and port 2, respectively. The green, red,
and black regions indicate self-induced oscillation, period-doubling
bifurcations, and chaos, respectively. (c),(d) Bifurcation diagrams of the
mechanical oscillation (upper panels) for pumping from port 1 and port 2 and
the corresponding maximal Lyapunov exponent spectra (bottom panels), with
$|\xi|/\Omega=3$. Dash-dotted lines mark $\lambda_{\rm max}=0$. Insets in (c)
present two types of phase-space trajectories of the light fields, i.e., chaos
and order. (e),(f) Chirality $C$ and symmetry $S$ between phase diagrams (a)
and (b). The black dash-dotted line in (f) marks $S=1$, and the three black
diamonds represent $\varphi=0$ and $\pm\;\\!\pi$. Parameters RMP861391 ; AOP7
: $\eta/\Omega=0.15,$ $J/\Omega=2,$ $\Gamma/\Omega=5\times 10^{-3},$
$\gamma/\Omega=5,$ $\kappa/\Omega=0.25,$ $G/\Omega=5\times 10^{-5},$
$\varepsilon/\Omega=5.8\times 10^{4},$ $(\omega_{\rm
A}\;\\!\\!-\;\;\\!\\!\\!\omega)/\Omega=(\omega_{\rm\;\;\\!\\!\\!B}\;\\!\\!-\;\;\\!\\!\\!\omega)/\Omega=$
$-\;\\!0.5$.

Chiral chaos and its metrics.—With a monochromatic pumping field with
frequency $\omega$ and amplitude $\varepsilon$ applied to either port, the
dynamics of the device can be obtained from the Heisenberg equations of motion
SupMat . The phase diagrams are presented in Figs. 2(a) and 2(b), where the
chaotic and ordered regimes are divided according to the maximal Lyapunov
exponent $\lambda_{\rm max}$ (i.e., chaos: $\lambda_{\rm max}>0$; order:
$\lambda_{\rm max}<0$) and power spectrum of the mechanical mode book2005 . It
is obvious that these two patterns are chiral along the
$\varphi\;\;\\!\\!\\!$-axis. With the continuous increase of $\varphi$ from
$-\;\\!\pi$ to $\pi$, the transition process between order state and chaos
state for pumping from port 2 is the reverse of that for pumping from port 1.
Furthermore, we note that the order-to-chaos transition can be controlled by
tuning the parameters $|\xi|$ and $\varphi$, which corresponds to adjusting
two tips in experiments, as shown in Fig. 1(c). The features depicted in two
phase diagrams indicate that the device supports a chiral chaos that varies
with $\varphi$. However, no similar chirality occurs on the $|\xi|$-axis. To
illustrate chiral chaos more intuitively, we draw the bifurcation diagrams and
the maximal Lyapunov exponent spectra in Figs. 2(c) and 2(d). The chirality
observed in the structure of two bifurcation diagrams, especially the
positions of periodic windows within the chaotic regime, indicates that chiral
chaos can be induced by tuning $\varphi$. The spectra of $\lambda_{\rm max}$
also verifies this conclusion.

Physically, the anomalous phenomena in Fig. 2 imply that time-reversal
symmetry of WGMs has been broken. Specifically, chirality results from the
couplings between optical modes, as shown in Fig. 1(b) and the Hamiltonian
$\hat{H}_{\rm eff}$ in Eq. (1). Photons hopping along two reverse loops
accumulate opposite net phases, i.e., hopping along loop $\hat{a}_{\rm
CCW}\rightarrow\hat{b}_{\rm CW}\rightarrow\hat{b}_{\rm
CCW}\rightarrow\hat{a}_{\rm CW}\rightarrow\hat{a}_{\rm CCW}$ and the reverse
loop $\;\\!\\!\hat{a}_{\rm CCW}\;\\!\\!\rightarrow\;\\!\\!\hat{a}_{\rm
CW}\;\\!\\!\rightarrow\;\\!\\!\hat{b}_{\rm CCW}\;\\!\\!\rightarrow\hat{b}_{\rm
CW}\;\\!\\!\rightarrow\hat{a}_{\rm CCW}\;\\!\\!$ accumulate phases
$-\;\\!\varphi$ and $\varphi$, respectively. By combining the gauge
transformation and time-reversal transformation PRA82043811 ; SupMat , we
conclude that once $\varphi\neq\mathbb{Z}\;\;\\!\\!\\!\pi$ (here, $\mathbb{Z}$
denotes arbitrary integer), the light fields in the device no longer hold
time-reversal symmetry. Similar to previous works PRL126123603 ; PRL122083903
, the net hopping phase marks the presence of a static optical gauge field,
which has nontrivial effects on the dynamics of the device. To illustrate
this, we define $\delta I_{\rm A}$ as the light intensity difference in
resonator A produced by inputting the same pumping field from two different
ports. For steady state, it can be expressed as $\delta I_{\rm
A}^{s}=g\;\;\\!\\!\\!(|\eta\;\;\\!\\!\\!+Fe^{i\varphi}|^{2}-|\eta\;\;\\!\\!\\!+Fe^{-i\varphi}|^{2})$
SupMat . We can find that for a trivial gauge phase, i.e.,
$\varphi=\mathbb{Z}\pi$, $\delta I_{\rm A}^{s}=0$, while for a nontrivial
gauge phase, the distribution of values of $\delta I_{\rm A}^{s}$ on the
$\varphi\;\;\\!\\!\\!$-axis exhibits chirality. Although the analytical
expression of $\delta I_{\rm A}$ for chaotic state is unobtainable, the
$\varphi$-dependent chirality for $\delta I_{\rm A}^{s}$ and $\delta I_{\rm
A}$ is the same as the chirality of chaos dynamics shown in Figs. 2(a) and
2(b).

To quantitatively characterize the degree of symmetry and chirality between
different routes to chaos, we design the following metrics based on the
Lyapunov exponents. For any two $N$-element conditional sets $\\{x_{i}\\}$ and
$\\{y_{i}\\}$, the system will produce equal-length state sets $\\{U_{i}\\}$
and $\\{V_{i}\\}$, $i=1,2,\;\;\\!\\!\\!\cdots\;\\!\\!,N$, respectively. By
applying efficient algorithms Meccanica1591980 ; Physicald , one can calculate
two corresponding maximal Lyapunov exponent sets $\\{\lambda_{\rm max}^{i}\\}$
and $\\{\Lambda_{\rm max}^{i}\\}$. Then we define a class of structural
parameters as

$\displaystyle S=$
$\displaystyle\;1-\dfrac{1}{N}\sum_{i=1}^{N}\dfrac{|\lambda_{\rm
max}^{i}-\Lambda_{\rm max}^{i}|}{|\lambda_{\rm max}^{i}|+|\Lambda_{\rm
max}^{i}|},$ (2) $\displaystyle C=$
$\displaystyle\;1-\dfrac{1}{N}\sum_{i=1}^{N}\dfrac{|\lambda_{\rm
max}^{i}-\Lambda^{N-i+1}_{\rm max}|}{|\lambda_{\rm
max}^{i}|+|\Lambda^{N-i+1}_{\rm max}|}.$ (3)

It is obvious that $S$ and $C$ reflect the degree of symmetry and chirality
between evolution processes $\\{U_{i}\\}$ and $\\{V_{i}\\}$. For the ideal
chirality and complete achirality, $C$ takes the values 1 and 0, respectively,
while for all other cases, $C\;\\!\\!\in\;\\!\\!(0,1)$. Similarly, $S$ will
reach the value of $1$ for complete symmetry and $0$ for complete asymmetry,
while for all other cases, $S\;\\!\\!\in\;\\!\\!(0,1)$. More discussions about
$S$ and $C$ are provided in Supplemental Material SupMat .

We now demonstrate the effectiveness of metrics $S$ and $C$. As shown in Fig.
2(e), the $\varphi\;\;\\!\\!\\!$-dependent chiral order-to-chaos transition
process (for each $|\xi|$,
$\varphi\;\\!\\!\in\;\\!\\![\;\;\\!\\!\\!-\;\\!\pi,\pi\;\;\\!\\!\\!]$) shows
strong consistency with the value of $C$. Moreover, the relation between $S$
and $|\xi|$ aligns well with the structural characteristics of phase diagrams.
In Fig. 2(f), the $|\xi|$-dependent order-to-chaos transition process (for
each $\varphi$,
$|\xi|/\Omega\in\;\\!\\![\;\;\\!\\!\\!2.5,\;\;\\!\\!\\!3.8\;\;\\!\\!\\!]$) of
two phase diagrams in Figs. 2(a) and 2(b) can also be accurately described by
$S$-curve and $C$-curve. It is also worth noting that $S$ and $C$ have great
distinguishing performance, as shown by the three black diamonds in Fig. 2(f),
where complete symmetry result in sharp rise of S.

Figure 3: (a) The maximum magnitude of the light intensity in resonator A
(i.e., $I_{\rm A,\;\\!max}=\;\;\\!\\!\\!{\rm max}\;\\!\\{|a_{\rm
CW}(\tau)|^{2}+\;\;\\!\\!\\!|a_{\rm CCW}(\tau)|^{2}\\}$) versus $\varphi$,
with $|\xi|/\Omega=3$. The red and blue curves represent the results of
pumping from port 1 and port 2, respectively. The sensing window within the
black dotted box is enlarged for convenient observation. The width of the
sensing window is determined by the difference of two phase transition points
induced by pumping from different ports, i.e., the width of the green strip.
(b) and (c) show the adjustability of the window half-width $D$ and center
position $P$, respectively. Point $F$ in (c) represents the working point of
the device. Other parameters are the same as in Fig. 2.

Chiral chaos for sensitive sensing.—We observe that a step-like rise or drop
in Figs. 2(c) and 2(d) marks the order-to-chaos phase transition, which can be
verified by power spectra of the light fields in resonators M.C.Gutzwiller .
Near the critical point, a small perturbation $\delta\varphi$ may cause
significant changes in the system states, as shown in the insets in Fig. 2(c)
and the process $\rm O\rightarrow M$ (orange solid arrow) in Fig. 2(d). Such a
transition has potential applications in high-precision sensing NatPhoton9151
; IEEE46440 . However, the orientation randomness of the unknown perturbation
$\delta\varphi$, i.e., positive deviation $\delta\varphi>0$ (denoted by
$+\;\;\\!\\!\\!\delta\varphi)$ or negative deviation
$\delta\varphi\;\\!\\!<\;\\!\\!0$ (denoted by $-\;\;\\!\\!\\!\delta\varphi)$,
inevitably results in false negative errors for conventional chaos sensing
without chirality during the sensing process, as shown in the $\rm
O\rightarrow O^{\prime}\;\\!\\!$ error (orange dotted arrow) in Fig. 2(d).
This error seriously limits the detection efficiency.

In our chiral chaotic device, the transitions of order-to-chaos and period-
doubling cascades no longer fully overlap for different pumping ports. As
shown in Fig. 3(a), the distribution of the light intensity from different
pumping ports satisfies chiral symmetry. Thus, we can synthesize several
stable sensing windows, where the detection of small signal can be realized by
splitting it into two which are alternatively switched to two ports. In this
case, the dynamics of the device is only sensitive to the strength
$|\delta\varphi|$ of the perturbation $\delta\varphi$ no matter what the
deviation orientation of $\delta\varphi$ is.

Performance of the sensing window mainly depends on its half-width $D$, center
position $P$, and working point $F$. We now investigate the performance of the
right sensing window in Fig. 3(a). The selection of the working point $F$ is
done according to the detection accuracy and is independent of $D$ and $P$.
However, as shown in Figs. 3(b) and 3(c), $D$ and $P$ strongly depend on the
parameters $J$, $|\xi|$ and $\varphi$ of the device. The parameter $J$ can be
adjusted via a translation stage Nature548192 , $|\xi|$ and $\varphi$ can be
tuned by tips as shown in Fig. 1(c), thus the synthesized sensing windows have
excellent flexibility. By comparing the effects of $|\xi|$ and $J$ on the
window in Figs. 3(b) and 3(c), we find that $J$ plays dominant role. This
indicates that the variation of the distance between resonators is suitable as
a coarse tuning method, while tips are suitable as fine adjustment knobs.
Based on the expected accuracy of the detection and chosen value of the half-
width $D$, the sensing windows and their applications can be further divided,
e.g., narrow windows with small $D$ values are applicable for high precision
sensing, wide windows with large $D$ values are applicable for threshold
alarms J.S.Wilson . Hereafter, we only discuss the narrow window case.

The above $\varphi$-dependent sensing can be generalized to other sensing
parameters. We now study sensing windows constructed on the
$(\varepsilon,\omega)$ plane of the pumping field SupMat . We assume that an
additional detected signal with frequency $\omega+\delta\omega$
($\delta\omega\ll\omega$) and amplitude $\delta\varepsilon$
($\delta\varepsilon\ll\varepsilon$) is applied to the device through the input
port. Thus, the total input field is $E_{\rm
tot}\;\\!\\!=\;\;\\!\\!\\!\varepsilon_{\rm tot}\;\;\\!\\!\\!{\rm
cos}\;\\!\;\\!\\!(\;\\!\;\\!\\!\omega\;\;\\!\\!\\!\tau+\theta_{\rm tot})$ with

$\displaystyle\begin{split}\varepsilon_{\rm
tot}&=\;\;\\!\\!\\!\sqrt{\varepsilon^{2}+\delta\varepsilon^{2}+2\;\;\\!\\!\\!\varepsilon\;\;\\!\\!\\!\delta\varepsilon\;\\!\\!\;\\!{\rm
cos}\;\;\\!\\!\\!(\delta\omega\;\;\\!\\!\\!\tau+\theta)}\;\;\\!\\!\\!,\\\\[3.0pt]
\theta_{\rm
tot}&=\;\;\\!\\!\\!\arctan\\!\left[\;\;\\!\\!\\!\dfrac{\delta\varepsilon\;\\!{\rm
sin}\;\;\\!\\!\\!(\delta\omega\;\;\\!\\!\\!\tau+\theta)}{\varepsilon+\delta\varepsilon\;\;\\!\\!\\!{\rm
cos}\;\;\\!\\!\\!(\delta\omega\;\;\\!\\!\\!\tau+\theta)}\;\\!\\!\;\\!\right]\\!\;\\!\\!.\end{split}$
(4)

$\varepsilon_{\rm tot}$ and $\theta_{\rm tot}$ are the equivalent amplitude
and additional phase, $\theta$ is the initial phase difference between the
pumping field and detected signal. To characterize the sensing performance, we
classify the sensing results into two categories, i.e., failure and success
based on following criteria: (i) chiral order-to-chaos phase transition is
induced, (ii) the amplitudes of physical quantities (e.g., light intensity and
mechanical oscillation amplitude) are significantly changed. If both criteria
are met after the detected signal is introduced, then we call it as a success,
otherwise as a failure.

Figure 4: ($\rm{a}_{1}$),($\rm{b}_{1}$) Results of conventional chaos-based
sensing proposal, obtained by inputting the pumping field and detected signal
from port 2. ($\rm{a}_{2}$)($\rm{a}_{3}$) and ($\rm{b}_{2}$) ($\rm{b}_{3}$)
Results of chiral-chaos-based sensing proposal using constructed sensing
windows. $\Delta I_{\rm A}$ is the change in the maximum light intensity in
resonator A before and after the detected signal is introduced. The yellow and
blue regions represent sensing failure and success, respectively. We summarize
the success rates of ($\rm{a}_{1}$)–($\rm{a}_{3}$) and
($\rm{b}_{1}$)–($\rm{b}_{3}$) in ($\rm{a}_{4}$) and ($\rm{b}_{4}$),
respectively, where the gray solid curves, orange dotted curves, and red
dashed curves correspond to ($\rm{a}_{1}$)($\rm{b}_{1}$),
($\rm{a}_{2}$)($\rm{b}_{2}$), and ($\rm{a}_{3}$)($\rm{b}_{3}$), respectively.
The success rate here is defined as the proportion of successful sensing
events to the total events, for weak signals with a fixed amplitude
$\delta\varepsilon$. Parameters: ($\rm{a}_{1}$)–($\rm{a}_{4}$)
$\delta\omega=0$, ($\rm{b}_{1}$)–($\rm{b}_{4}$) $\theta/\pi=0.5$,
($\rm{a}_{1}$)($\rm{a}_{2}$) and ($\rm{b}_{1}$)($\rm{b}_{2}$)
$\varphi/\pi=0.505$, ($\rm{a}_{3}$) and ($\rm{b}_{3}$) $\varphi/\pi=0.5$. For
more details, see Supplemental Material SupMat .

We first consider the special case that the detected signal is in resonance
with the pumping field, i.e., $\delta\omega=0$, and the results are shown in
Figs. 4($\rm{a}_{1}$)–($\rm{a}_{4}$). For a conventional sensing using order-
to-chaos phase transition IEEE46440 , as shown in Fig. 4($\rm{a}_{1}$), its
performance is strongly affected by the value of $\theta$, however $\theta$
cannot be known a priori in the actual sensing process, which will inevitably
lead to failure in some cases. Thus, the success rate is below 50% as shown in
the gray solid curve in Fig. 4($\rm{a}_{4}$). In contrast to conventional one,
the chiral chaos based sensing exhibits stronger robustness to $\theta$ as
shown in Fig. 4($\rm{a}_{2}$) for given $\varphi$, e.g., $\varphi/\pi=0.505$,
and the corresponding success rate is also significantly enhanced as shown in
the orange dotted curve in Fig. 4($\rm{a}_{4}$). By further optimizing
$\varphi$, we can even achieve sensing that is immune to $\theta$, and the
success rate can approach 100%, as shown in Fig. 4($\rm{a}_{3}$) and the red
dashed curve in Fig. 4($\rm{a}_{4}$) for, e.g., $\varphi/\pi=0.5$.

For $\delta\omega\neq 0$, the effective detectable range of $\delta\omega$,
defined as continuously range of $\delta\omega$ for sensing success, is an
important index to evaluate the quality of a sensor. Fig. 4($\rm{b}_{1}$)
shows a serious flaw of conventional proposal, i.e., effective detectable
range is very narrow, only a few $\rm MHz$. However, our proposal utilizing
chiral chaos has great advantages in this aspect, as shown in Fig.
4($\rm{b}_{2}$) for given $\varphi$, e.g., $\varphi/\pi=0.505$, an effective
detectable range with tens of $\rm MHz$ is achieved. By modulating the chiral
chaos by optimizing $\varphi$, the effective detectable range can exceed 100
$\rm MHz$ as shown in Fig. 4($\rm{b}_{3}$). Besides, the success rates in Fig.
4($\rm{b}_{4}$) also prove the superiority of our proposal. In practice, the
effective detectable range is also affected by the bandwidth of the pumping
field.

We further analyze these results. As shown in Eq. (4), if $\delta\omega=0$,
$\theta$ controls the enhancement ($\varepsilon_{\rm tot}>\varepsilon$) or
attenuation ($\varepsilon_{\rm tot}<\varepsilon$) of the pumping field. This
implies that the upper limit of the success rate is only 50% for conventional
proposal, since the order-chaos phase transition can only follow one route no
matter how the signal is applied. However, the situation is changed through
chiral chaos. By splitting the signal into two which are alternatively
switched to two ports, order-chaos phase transition through a sensing window
can proceed along two opposite routes SupMat , which ensure that the
theoretical upper limit of the success rate is 100%.

For $\delta\omega\neq 0$, $\theta$ mainly affects the initial conditions. The
modulation of the pumping field by the detected signal is nonmonotonic,
$\varepsilon_{\rm tot}$ is periodically less or more than $\varepsilon$, and
$\theta_{\rm tot}$ also varies periodically. The system undergoes chaos-to-
order transition within half the time, while the remaining half is dedicated
to maintaining the current state. The final effect of the competition process
depends on the skeleton of the chaotic attractors and the initial conditions
book2004 . For chiral chaos, the structures of attractors corresponding to
different input directions are inconsistent, resulting in more attractors
gathered in the region of the sensing window than those of conventional chaos.
Thus, the effective detection range for our chiral-chaos-based sensing is
wider than that of convention one.

Discussion and conclusion.—In summary, we propose chiral chaos and construct a
chiral chaotic device. The phase diagrams of chiral chaos and chaotic dynamics
are carefully investigated. Moreover, we introduce two structure parameters as
metrics of symmetry and chirality. Based on chiral chaos, we show that the
false negative errors plaguing conventional chaos-based sensing proposals
IEEE46440 ; NatPhoton9151  can be partially or even completely overcome. In
our study for chiral-chaos-based sensing, the required quality factor
$Q\in[10^{7},10^{8}]$ for the WGM resonators is achievable NatPhoton4462010 ;
PRL104083901 . The controllable coupling parameters can be realized via the
translation stages OE1823535 ; AOP7 . Moreover, these parameters can be
further optimized by changing the medium environment or electroplating
specific materials on tips Nature548192 . We finally mention that our proposal
can also be generalized to other physical systems, e.g., superconducting
quantum circuits PhysRep.718.1  and systems of quantum acoustics Shao-yang .

Acknowledgments.—Y.X.L. is supported by the National Natural Science
Foundation of China with Grants No. 12374483 and No. 92365209. J.Z. is
supported by Tsinghua-Foshan Innovation Special Fund (20182000306); Department
of Liaoning Province and State Key Laboratory of Robotics, China
(2021-KF-22-01); Wenhai Program of the S&T Fund of Shandong Province for Pilot
National Laboratory for Marine Science and Technology (Qingdao)
(NO.2021WHZZB2500, NO.2021WHZZB2501).

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Supplemental Material: Chiral Chaos Enhanced Sensing

##  I Details of the chiral chaotic device presented in the main text

In a rotating frame at the pumping field frequency $\omega$, the Hamiltonian
of our system can be written as

$\displaystyle\hat{H}=$ $\displaystyle\ \hbar\;\;\\!\\!\\!\Delta_{\rm
A}(\hat{a}^{{\dagger}}_{\rm CW}\hat{a}_{\rm CW}+\hat{a}^{{\dagger}}_{\rm
CCW}\hat{a}_{\rm
CCW})+\frac{\hbar}{2}\;\;\\!\\!\\!\Omega\;\;\\!\\!\\!(\;\;\\!\\!\\!\hat{p}^{2}+\hat{q}^{2})-\hbar\;\;\\!\\!\\!G(\hat{a}^{{\dagger}}_{\rm
CW}\hat{a}_{\rm CW}+\hat{a}^{{\dagger}}_{\rm CCW}\hat{a}_{\rm
CCW})\hat{q}+\hbar\;\\!\\!\\!\;\Delta_{\rm B}(\hat{b}^{{\dagger}}_{\rm
CW}\hat{b}_{\rm CW}+\hat{b}^{{\dagger}}_{\rm CCW}\hat{b}_{\rm CCW})$
$\displaystyle-\hbar\;\;\\!\\!\\!J(\hat{a}^{{\dagger}}_{\rm CW}\hat{b}_{\rm
CCW}+\hat{a}^{{\dagger}}_{\rm CCW}\hat{b}_{\rm CW}+\hat{a}_{\rm
CW}\hat{b}^{{\dagger}}_{\rm CCW}+\hat{a}_{\rm CCW}\hat{b}^{{\dagger}}_{\rm
CW})-\hbar\;\;\\!\\!\\!\eta\;\;\\!\\!\\!(\hat{a}^{{\dagger}}_{\rm
CW}\hat{a}_{\rm CCW}+\hat{a}_{\rm CW}\hat{a}^{{\dagger}}_{\rm CCW})$
$\displaystyle-\hbar\;\;\\!\\!\\!|\xi|(e^{i\varphi}\hat{b}^{{\dagger}}_{\rm
CW}\hat{b}_{\rm CCW}+e^{-i\varphi}\hat{b}_{\rm CW}\hat{b}^{{\dagger}}_{\rm
CCW})+i\hbar\;\;\\!\\!\\!\varepsilon_{1}(\hat{a}_{\rm
CW}^{{\dagger}}-\hat{a}_{\rm
CW})+i\hbar\;\;\\!\\!\\!\varepsilon_{2}(\hat{a}_{\rm
CCW}^{{\dagger}}-\hat{a}_{\rm CCW}).$ (S1)

Dynamics of the above hybrid optomechanical system are described by the
Heisenberg equations of motion

$\displaystyle\dot{a}_{\rm CCW}$ $\displaystyle=-i(\Delta_{\rm
A}-i\kappa)a_{\rm CCW}+iGa_{\rm CCW}q+i\eta\;\;\\!\\!\\!a_{\rm CW}+iJb_{\rm
CW}+\varepsilon_{2},$ (S2) $\displaystyle\dot{a}_{\rm CW}$
$\displaystyle=-i(\Delta_{\rm A}-i\kappa)a_{\rm CW}+iGa_{\rm
CW}q+i\eta\;\;\\!\\!\\!a_{\rm CCW}+iJb_{\rm CCW}+\varepsilon_{1},$ (S3)
$\displaystyle\dot{b}_{\rm CCW}$ $\displaystyle=-i(\Delta_{\rm
B}-i\gamma)b_{\rm CCW}+i|\xi|e^{-i\varphi}b_{\rm CW}+iJa_{\rm CW},$ (S4)
$\displaystyle\dot{b}_{\rm CW}$ $\displaystyle=-i(\Delta_{\rm
B}-i\gamma)b_{\rm CW}+i|\xi|e^{i\varphi}b_{\rm CCW}+iJa_{\rm CCW},$ (S5)
$\displaystyle\dot{q}$
$\displaystyle=\Omega\>\\!p,\quad\dot{p}=-\;\\!\Omega\>\\!q+G(|a_{\rm
CW}|^{2}+|a_{\rm CCW}|^{2})-\Gamma\>\\!p,$ (S6)

where $a_{\rm CW}=\langle\hat{a}_{\rm CW}\rangle$ and $a_{\rm
CCW}=\langle\hat{a}_{\rm CCW}\rangle$ ($b_{\rm CW}=\langle\hat{b}_{\rm
CW}\rangle$ and $b_{\rm CCW}=\langle\hat{b}_{\rm CCW}\rangle$) represent the
amplitudes of CW and CCW modes in resonator A (B),
$q=\langle\hat{q}\;\;\\!\\!\\!\rangle$ and
$p=\langle\;\;\\!\\!\\!\hat{p}\;\;\\!\\!\\!\rangle$ are the displacement and
momentum of the mechanical mode (for any operator $\hat{o}$,
$o\equiv\langle\hat{o}\;\;\\!\\!\\!\rangle$ denotes its expectation value).
Here, correlations between optical modes and mechanical mode have been ignored
S (1, 2). $\Delta_{\rm A}=\;\;\\!\\!\\!\omega_{\rm A}\;\\!\\!-\omega\
(\Delta_{\rm B}=\;\;\\!\\!\\!\omega_{\rm\;\;\\!\\!\\!B}-\omega)$ denotes the
detuning between WGMs in resonator A (B) and the pumping field, $\kappa$
$(\gamma\;\;\\!\\!\\!)$ is the damping rate of WGMs in resonator A (B),
$\varepsilon_{j}$ denotes the amplitude of the pumping field at port $j$
($\;\;\\!\\!\\!j=1,2$), $\Omega$ and $\Gamma$ are the mechanical resonance
frequency and damping rate, $G$ is the single-photon optomechanical coupling
strength, $\eta$ and $|\xi|e^{\pm i\varphi}$ are the backscattering strengths
of WGMs in resonators A and B, respectively, and $J$ is the tunneling strength
between resonators. We now provide more details on it.

###  I.1 Tips-induced coupling between CW and CCW modes under the dipole
approximation

The electric field distribution in resonator B is

$\displaystyle\hat{\bm{E}}(\bm{r},t)=f(\bm{r})\sqrt{\frac{\hbar\;\\!\omega_{\rm
B}}{2\;\;\\!\\!\\!\varepsilon_{0}V_{\rm B}}}\;\;\\!\\!\\!(\hat{b}_{\rm
CW}e^{-i\;\;\\!\\!\\!\omega_{\rm B}t+i\bm{k}\cdot\bm{r}}+\hat{b}_{\rm
CCW}e^{-i\;\;\\!\\!\\!\omega_{\rm B}t-i\bm{k}\cdot\bm{r}}+{\rm
H.c.})\;\\!\;\\!\\!\bm{e}(\bm{r}),$ (S7)

where $\omega_{\rm\;\;\\!\\!\\!B}$ and $V_{\rm B}$ denote the resonant
frequency and quantization volume of WGMs in resonator B, $\varepsilon_{0}$ is
the vacuum permittivity, $\hat{b}_{\rm CW}$ and $\hat{b}^{\dagger}_{\rm CW}$
($\hat{b}_{\rm CCW}$ and $\hat{b}^{\dagger}_{\rm CCW}$) represent the
annihilation and creation operators of CW (CCW) mode. Here, $\bm{k}$
$\;\\!\\!(-\bm{k})\;\\!\\!$ denotes the wave vector of CW (CCW) traveling wave
mode, $\bm{e}(\bm{r})\;\\!\\!$ and $\\!f(\bm{r})\;\\!\\!$ represent the unit
vector of electric field direction and cavity-mode function at position
$\bm{r}$. Two tips contact the rim of resonator B at $\bm{r}_{1}$ and
$\bm{r}_{2},$ causing them to be polarized by the evanescent fields, which can
be described as

$\displaystyle\hat{\bm{P}}(\bm{r}_{1},t)=\varepsilon_{0}\;\;\\!\\!\\!\chi_{1}\hat{\bm{E}}(\bm{r}_{1},t),\quad\hat{\bm{P}}(\bm{r}_{2},t)=\varepsilon_{0}\;\;\\!\\!\\!\chi_{2}\;\;\\!\\!\\!\hat{\bm{E}}(\bm{r}_{2},t),$
(S8)

in which $\chi_{1}$ and $\chi_{2}$ represent the polarizability of tip $1$ and
tip $2$. Utilizing the dipole approximation and rotating wave approximation,
and considering the backscattering process with strength $\xi_{0}$ caused by
surface defects, the interaction energy between CW and CCW modes can be
obtained as S (3)

$\displaystyle\hat{V}=$
$\displaystyle-\frac{1}{2}\;\;\\!\\!\\!\hat{\bm{P}}(\bm{r}_{1},t)\;\\!\\!\cdot\\!\hat{\bm{E}}(\bm{r}_{1},t)-\frac{1}{2}\;\;\\!\\!\\!\hat{\bm{P}}(\bm{r}_{2},t)\;\\!\\!\cdot\\!\hat{\bm{E}}(\bm{r}_{2},t)-\hbar\;\\!\xi_{0}(\hat{b}_{\rm
CW}^{{\dagger}}\hat{b}_{\rm CCW}+\hat{b}_{\rm CW}\hat{b}^{{\dagger}}_{\rm
CCW})$ $\displaystyle=$ $\displaystyle-\hbar\;\\!\delta(\hat{b}_{\rm
CW}^{{\dagger}}\hat{b}_{\rm CW}+\hat{b}^{{\dagger}}_{\rm CCW}\hat{b}_{\rm
CCW})-\hbar\;\;\\!\\!\\!|\xi|(e^{i\varphi}\hat{b}^{{\dagger}}_{\rm
CW}\hat{b}_{\rm CCW}+e^{-i\varphi}\hat{b}_{\rm CW}\hat{b}^{{\dagger}}_{\rm
CCW}),$ (S9)

in which

$\displaystyle\delta=\frac{\omega_{\rm
B}(\chi_{1}f^{2}(\bm{r}_{1})+\chi_{2}f^{2}(\bm{r}_{2}))}{2V_{\rm
B}},\quad|\xi|e^{\pm i\varphi}=\xi_{0}+\frac{\omega_{\rm
B}(\chi_{1}f^{2}(\bm{r}_{1})+\chi_{2}f^{2}(\bm{r}_{2})e^{\pm
i2n\beta})}{2V_{\rm B}}.$ (S10)

Note that the dipole-dipole interaction has been ignored owing to the
sufficient distance between tips. Without loss of generality, we have chosen
$\bm{r}_{1}$ as the origin of coordinates. $n$ is the azimuthal order
corresponding to the wave vector $\bm{k}$, i.e., the number of wavelengths in
the perimeter of resonator B, $\beta$ is the spatial phase difference between
two tips, and $\delta$ is the frequency shift of the optical modes induced by
two tips, according to the common experiment conditions S (4, 5), we calculate
its effective range in Fig. S1(a).

###  I.2 Tips-induced broadening of CW and CCW modes under the Weisskopf-
Wigner approximation

In addition to inducing coupling between optical modes, tips also lead to
additional linewidth broadening. The Hamiltonian of the WGMs coupled to the
reservoir modes can be expressed as

$\displaystyle\hat{H}=\;\;\\!\\!\\!\hbar\;\\!\omega_{\rm\;\;\\!\\!\\!B}(\hat{b}^{{\dagger}}_{\rm
CW}\hat{b}_{\rm CW}+\hat{b}^{{\dagger}}_{\rm CCW}\hat{b}_{\rm
CCW})+\sum_{s}\hbar\;\\!\omega_{s}\hat{c}^{{\dagger}}_{s}\hat{c}_{s}-\sum_{l}\sum_{s}\hbar\;\\!\;\\!\\!g_{l,s}(\hat{b}_{\rm
CW}^{{\dagger}}\hat{c}_{s}e^{i(\bm{k}_{s}-\bm{k})\cdot\bm{r}_{l}}+\hat{b}_{\rm
CCW}^{{\dagger}}\hat{c}_{s}e^{i(\bm{k}_{s}+\bm{k})\cdot\bm{r}_{l}}+\rm H.c.),$
(S11)

where $\hat{c}_{s}$ and $\hat{c}^{{\dagger}}_{s}$ are the annihilation and
creation operators of the $s\rm th$ reservoir mode with frequency
$\omega_{s}$, respectively.
$g_{l,s}=\chi_{l}f(\bm{r}_{l})\sqrt{\omega_{\rm\;\;\\!\\!\\!B}\;\\!\omega_{s}/4V_{\rm
B}V_{\rm vac}}\;\\!\bm{e}_{l}\\!\cdot\\!\bm{e}_{s}$ denotes the coupling
strength between the optical modes and the $s\rm th$ reservoir mode induced by
tip $l$ positioned at $\bm{r}_{l}$, where $\bm{e}_{s}$ and $V_{\rm vac}$
denote the unit vector and quantization volume of the $s\rm th$ reservoir
mode, and $\bm{e}_{l}$ denotes the unit vector of the WGMs. Then the
Heisenberg-Langevin equations become

$\displaystyle\dot{\hat{b}}_{\rm
CW}=-i\;\;\\!\\!\\!\omega_{\rm\;\;\\!\\!\\!B}\hat{b}_{\rm
CW}+i\sum_{l}\sum_{s}g_{l,s}\;\;\\!\\!\\!\hat{c}_{s}e^{i(\bm{k}_{s}-\bm{k})\cdot\bm{r}_{l}},$
(S12) $\displaystyle\dot{\hat{b}}_{\rm
CCW}=-i\;\;\\!\\!\\!\omega_{\rm\;\;\\!\\!\\!B}\hat{b}_{\rm
CCW}+i\sum_{l}\sum_{s}g_{l,s}\;\;\\!\\!\\!\hat{c}_{s}e^{i(\bm{k}_{s}+\bm{k})\cdot\bm{r}_{l}},$
(S13)
$\displaystyle\dot{\hat{c}}_{s}=-i\;\;\\!\\!\\!\omega_{s}\hat{c}_{s}+i\sum_{l}g_{l,s}(\hat{b}_{\rm
CW}e^{-i(\bm{k}_{s}-\bm{k})\cdot\bm{r}_{l}}+\hat{b}_{\rm
CCW}e^{-i(\bm{k}_{s}+\bm{k})\cdot\bm{r}_{l}}\;\\!\\!).$ (S14)

Further processing Eq. (S14) by formal integration gives

$\displaystyle\hat{c}_{s}(t)=\hat{c}_{s}(0)e^{-i\;\;\\!\\!\\!\omega_{s}t}+i\sum_{l}g_{l,s}\\!\int_{0}^{t}\\!\big{[}\hat{b}_{\rm
CW}(t^{\prime})e^{-i(\bm{k}_{s}-\bm{k})\cdot\bm{r}_{l}}+\hat{b}_{\rm
CCW}(t^{\prime})e^{-i(\bm{k}_{s}+\bm{k})\cdot\bm{r}_{l}}\big{]}e^{-i\;\;\\!\\!\\!\omega_{s}(t-t^{\prime})}dt^{\prime}.$
(S15)

We now separate the slowly varying and fast oscillatory contributions of CW
and CCW modes

$\displaystyle\hat{b}_{\rm CW}(t)=\tilde{b}_{\rm
CW}(t)e^{-i\;\;\\!\\!\\!\omega_{\rm B}t},\quad\hat{b}_{\rm
CCW}(t)=\tilde{b}_{\rm CCW}(t)e^{-i\;\;\\!\\!\\!\omega_{\rm B}t}.$ (S16)

Then Eqs. (S12) and (S13) can be rewritten as

$\displaystyle\dot{\tilde{b}}_{\rm
CW}(t)=-\sum_{l}\sum_{s}g_{l,s}^{2}\\!\int_{0}^{t}\\!\big{[}\tilde{b}_{\rm
CW}(t^{\prime})+\tilde{b}_{\rm
CCW}(t^{\prime})e^{-i2\bm{k}\cdot\bm{r}_{l}}\big{]}e^{-i(\omega_{s}-\omega_{\rm
B})(t-t^{\prime})}dt^{\prime}+i\sum_{l}\sum_{s}g_{l,s}\;\;\\!\\!\\!\hat{c}_{s}(0)e^{-i(\omega_{s}-\omega_{\rm
B})t}e^{i(\bm{k}_{s}-\bm{k})\cdot\bm{r}_{l}},$ (S17)
$\displaystyle\dot{\tilde{b}}_{\rm
CCW}(t)=-\sum_{l}\sum_{s}g_{l,s}^{2}\\!\int_{0}^{t}\\!\big{[}\tilde{b}_{\rm
CCW}(t^{\prime})+\tilde{b}_{\rm
CW}(t^{\prime})e^{i2\bm{k}\cdot\bm{r}_{l}}\big{]}e^{-i(\omega_{s}-\omega_{\rm
B})(t-t^{\prime})}dt^{\prime}+i\sum_{l}\sum_{s}g_{l,s}\;\;\\!\\!\\!\hat{c}_{s}(0)e^{-i(\omega_{s}-\omega_{\rm
B})t}e^{i(\bm{k}_{s}+\bm{k})\cdot\bm{r}_{l}},$ (S18)

where the coupling terms between the CW and CCW modes at different positions
caused by the reservoir have been neglected. The fluctuation operators
associated with $\hat{c}_{s}(0)$ will be ignored in the following, since they
have no effect on the mean value equations ($\langle\hat{c}_{s}(0)\rangle=0$).
If we further assume that the reservoir modes are closely spaced in frequency,
then the following substitution relation holds

$\displaystyle\sum_{s}\;\rightarrow\;\frac{2V_{\rm vac}}{(2\pi
v)^{3}}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}{\rm
sin}(\theta)d\theta\int_{0}^{\infty}\omega_{s}^{2}d\omega_{s},$ (S19)

in which we have set the optical fields to be polarized along the $z$-axis,
$\theta$ and $\phi$ describe the polarization direction of the reservoir field
in spherical coordinate system, and $v$ accounts for the velocity of light in
the medium surrounding resonator B. Combining Eqs. (S17)–(S19), the tips-
induced broadening of CW and CCW modes can be obtained as

$\displaystyle\dot{\tilde{b}}_{\rm CW}(t)=-\;\\!\gamma_{\rm t}\tilde{b}_{\rm
CW}(t)-|\zeta|e^{i\Theta}\tilde{b}_{\rm CCW}(t),\quad\dot{\tilde{b}}_{\rm
CCW}(t)=-\;\\!\gamma_{\rm t}\tilde{b}_{\rm
CCW}(t)-|\zeta|e^{-i\Theta}\tilde{b}_{\rm CW}(t),$ (S20)

in which

$\displaystyle\gamma_{\rm t}=\frac{\omega_{\rm
B}^{4}(\chi_{1}^{2}f^{2}(\bm{r}_{1})+\chi_{2}^{2}f^{2}(\bm{r}_{2}))}{12\pi
v^{3}V_{\rm B}},\quad|\zeta|e^{\pm i\Theta}=\frac{\omega_{\rm
B}^{4}(\chi_{1}^{2}f^{2}(\bm{r}_{1})+\chi_{2}^{2}f^{2}(\bm{r}_{2})e^{\pm
i2n\beta})}{12\pi v^{3}V_{\rm B}}.$ (S21)

If we further ignore the shape details of tips and treat them as spheres, then
the polarizability of tips can be expressed as $\chi_{l}=4\;\;\\!\\!\\!\pi
R_{l}^{\;\;\\!\\!\\!3}(n_{l}^{2}-1)/(n_{l}^{2}+2)$, where $R_{l}$ and $n_{l}$
denote the effective radius and refractive index of the sphere scatterer
equivalent by tip $l$. After combining specific parameters, we calculate the
value ranges of $|\zeta|e^{\pm i\Theta}$ and $\gamma_{\rm t}$, as shown in
Figs. S1(b) and S1(c). Furthermore, Eqs. (S10) and (S21) also state that
$\delta$ and $|\xi|e^{\pm i\varphi}$ are proportional to
$R_{l}^{\;\;\\!\\!\\!3}$, while $\gamma_{\rm t}$ and $|\zeta|e^{\pm i\Theta}$
are proportional to $R_{l}^{\;\;\\!\\!\\!6}$, thus, for small scale tips
$\\{|\xi|e^{\pm i\varphi},\delta\\}\gg\\{|\zeta|e^{\pm i\Theta},\gamma_{\rm
t}\\}$.

Figure S1: (a) $\delta$ as a function of the geometric scale (effective
radius) of two tips. (b) $|\zeta|e^{\pm i\Theta}$ as a function of the
geometric scale and relative position of two tips, where the green and gray
regions represent the available and unavailable ranges. (c) $\gamma_{\rm t}$
as a function of the geometric scale (effective radius) of two tips. In this
Letter, the following parameters are used in all figures,
$n_{1}^{2}=n_{2}^{2}=3.9$, $\Omega=2\pi\times 20$ $\rm MHz$,
$f(\bm{r}_{1})=f(\bm{r}_{2})=0.3$, $V_{\rm B}=200\;\rm\mu m^{3}$, $v=3\times
10^{8}\;\rm m/s$, $\omega_{\rm\;\;\\!\\!\\!B}=2\pi\times 190\;\rm THz$,
$\beta\in[\;\;\\!\\!\\!-\;\\!\pi,\pi\;\;\\!\\!\\!]$, $R_{1}$ and
$R_{2}\in[\;\;\\!\\!\\!20,80\;\;\\!\\!\\!]\;\rm nm$.

##  II Details of chiral chaos and its metrics

We numerically solve the equations of motion Eqs. (S2)–(S6) to obtain the
dynamics of the hybrid optomechanical system. To improve the accuracy of
calculations and simplify the discussions, we normalize all frequency scales
to the mechanical oscillation frequency $\Omega$ and extend the time scale to
$\tau=\Omega\;\;\\!\\!\\!t$. Furthermore, we discard the initial transient
state $\tau_{0}/2\pi\in[\;\;\\!\\!\\!0,8000\\!\\!\\!\;\;]$ in all figures. It
should be noted that linear and nonlinear interaction terms introduce enormous
complexity to the equations of motion, which may lead to the coexistence of
different limit cycles in certain regions of parameter space S (6). In order
to reduce its interference to subsequent discussions, the initial conditions
we adopted in all numerical calculations are $a_{\rm CW}(0)=a_{\rm
CCW}(0)=b_{\rm CW}(0)=b_{\rm CCW}(0)=q(0)=p(0)=0$.

###  II.1 Main operating modes of the chiral chaotic device

Figure S2: (a)–(d) Time-domain evolutions of the total light intensity in
resonator A $(I_{\rm A}=|a_{\rm CW}|^{2}+|a_{\rm CCW}|^{2})$ for pumping from
port 1 (red) and port 2 (blue), under different parameter conditions. Here,
the hopping phases in the two regions demarcated by the green and gray dotted
lines are $\varphi/\pi=0.755$ and $\varphi/\pi=-\;\\!0.755$, respectively.
(e)–(h) The corresponding phase trajectories. (i)–(l) The corresponding
frequency spectra. Considering the current fabrication capabilities of optical
microresonators S (8, 9), other parameters chosen here are $\Omega=2\pi\times
20\;\rm MHz$, $G/\Omega=5\times 10^{-5},$ $\Gamma/\Omega=5\times 10^{-3},$
$\Delta_{\rm A}/\Omega=-\;\\!0.5,$ $\Delta_{\rm B}/\Omega=-\;\\!0.5$,
$\eta/\Omega=0.15$, $J/\Omega=2$, $|\xi|/\Omega=3.29$, $\kappa/\Omega=0.25$,
$\gamma/\Omega=5$, and $\varepsilon/\Omega=5.8\times 10^{\;\;\\!\\!\\!4}$.

Experimentally, the operating mode of our device can be detected by the
temporal evolution, power spectrum, and phase diagram of the intracavity light
fields S (7). Therefore, we present them all in Fig. S2. As shown in Figs.
S2(a), S2(e), and S2(i), when pumping from port 1, $I_{\rm A}$ exhibits
significant irregularities in the time domain, and its motion trajectories are
intertwined, forming a series of solid ribbon structures on the phase plane,
and its frequency spectrum appears continuous. All of these phenomena indicate
that the device is operating in chaotic mode, which means that even if the
equations of motion are deterministic, we still cannot accurately predict the
future state of the device.

However, if we switch the pumping port from 1 to 2 and keep other parameters
constant, then the operating mode of the device will change dramatically. As
shown in Figs. S2(b) and S2(j), $I_{\rm A}$ is in a stable periodic
oscillation state, which can be seen more intuitively from the frequency
spectrum (discrete peaks occur only at integer multiples of $\Omega$).
Although the motion trajectories in phase space appear somewhat disorganized
[see Fig. S2(f)], the obvious limit-cycle characteristics can still be
discerned, which also proves that the device is operating in an ordered mode.
In summary, the Lorentz reciprocity of our device has been broken (except for
some special cases, which will be discussed in detail later), and the same
conclusion can be drawn from Figs. S2(c), S2(d), S2(g), S2(h), S2(k), and
S2(l).

Figure S3: Bifurcation diagrams of the mechanical oscillation amplitude for
pumping from port 1 (red) and port 2 (blue), under different parameter
conditions. Here, for panels (a) and (b), $|\xi|/\Omega=3$ and both insets are
enlarged views; for panel (c), $\Delta=\omega_{\rm A}-\omega=\omega_{\rm
B}-\omega$, $\varepsilon/\Omega=5.4\times 10^{\;\;\\!\\!\\!4}$,
$|\xi|/\Omega=5$, and $\varphi/\pi=0.5$; for panel (d), $\varphi/\pi=0.5$ and
$|\xi|/\Omega=10$; other parameters in all panels are the same as in Fig. S2.

Another important finding is that if the hopping phase is transformed from
$\varphi$ to $-\;\\!\varphi$ while switching the pumping port, then the
operating state of the device remains unchanged (compare the left and right
parts of Fig. S2). To fully demonstrate this, in Figs. S3(a) and S3(b), we
plot bifurcation diagrams (vs $\varphi$) under two pumping conditions. Now, we
just need to count the number of bifurcations to distinguish different
operating modes S (10). More specifically, one, multiple, and infinite scatter
points on the diagram indicate the self-induced oscillation, period-doubling,
and chaotic modes of the device, respectively. A remarkable phenomenon is that
no matter which port is pumped, the device can be switched between regular and
chaotic operating modes by adjusting $\varphi$. However, such switching
processes are not completely independent for different pumping ports.
Comparing the results in Figs. S3(a) and S3(b), it can be easily found that
they satisfy chiral symmetry on the $\varphi\;\\!\;\\!\\!$-$\rm axis$, which
is particularly evident on the two insets. Therefore, we conclude that the
proposed device supports a chiral chaotic operating mode.

In addition, it is also worth noting that in certain parameter regions, chaos
can only be generated along one specific pumping direction. To show this
nonreciprocal chaotic operating mode more clearly, we plot the representative
results in Figs. S3(c) and S3(d). Both figures demonstrate that the response
of our device to external pumping (frequency $\omega$ and amplitude
$\varepsilon$) depends on the operating port. As shown in the red parts of
Figs. S3(c) and S3(d), when pumping from port 1, adjusting $\omega$ or
$\varepsilon$ can control the device to operate in a desired mode, either
chaotic or regular. However, after switching the pumping port, as shown in the
blue parts of Figs. S3(c) and S3(d), device will be restricted to the regular
operating regime, which means that all chaotic modes are no longer available.
Similar nonreciprocal behavior exists for other parameter regions as well.
Thus, the designed device can be applied to realize multiple types of
nonreciprocal signal modulation, which indicates its potential applications in
communication networks and quantum information processing S (11, 12).

###  II.2 Analysis for gauge transformations and time-reversal symmetry

Physically, the chirality in our system originates from the manipulating of
degeneracy or time-reversal symmetry of CW and CCW modes by tips. We now
demonstrate this from the perspective of gauge transformations. According to
the standard analysis method, if our system maintains time-reversal symmetry,
then its Hamiltonian needs to satisfy the transformation relation
$\hat{\Theta}\hat{H}\hat{\Theta}^{-1}=\hat{H}$, where $\hat{\Theta}$
represents the antiunitary and antilinear time-reversal operation.
Furthermore, the transformation relations of optical and mechanical operators
under time-reversal reads

$\displaystyle\hat{\Theta}\hat{a}_{\rm
CW}\hat{\Theta}^{-1}=e^{i\theta_{1}}\hat{a}_{\rm
CW},\quad\hat{\Theta}\hat{a}_{\rm
CCW}\hat{\Theta}^{-1}=e^{i\theta_{2}}\hat{a}_{\rm
CCW},\quad\hat{\Theta}\hat{q}\;\\!\;\\!\\!\hat{\Theta}^{-1}=\hat{q},$ (S22)
$\displaystyle\hat{\Theta}\hat{b}_{\rm
CW}\hat{\Theta}^{-1}=e^{i\theta_{3}}\hat{b}_{\rm
CW},\quad\hat{\Theta}\hat{b}_{\rm
CCW}\hat{\Theta}^{-1}=e^{i\theta_{4}}\hat{b}_{\rm
CCW},\quad\hat{\Theta}\hat{p}\;\\!\;\\!\\!\hat{\Theta}^{-1}=-\;\;\\!\\!\\!\hat{p}+\nabla\vartheta(\hat{q}),$
(S23)

where $\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4},\vartheta(\hat{q})\\}$
represent the linear gauge phases. One can easily verify that by virtue of the
gauge phases, the following relations always hold

$\displaystyle\hat{\Theta}(\hat{a}^{{\dagger}}_{\rm CW}\hat{a}_{\rm
CW}+\hat{a}^{{\dagger}}_{\rm CCW}\hat{a}_{\rm
CCW})\hat{q}\;\;\\!\\!\\!\hat{\Theta}^{-1}$
$\displaystyle=(\hat{a}^{{\dagger}}_{\rm CW}\hat{a}_{\rm
CW}+\hat{a}^{{\dagger}}_{\rm CCW}\hat{a}_{\rm CCW})\hat{q},$ (S24)
$\displaystyle\hat{\Theta}(\hat{a}^{{\dagger}}_{\rm CW}\hat{a}_{\rm
CW}+\hat{a}^{{\dagger}}_{\rm CCW}\hat{a}_{\rm CCW})\hat{\Theta}^{-1}$
$\displaystyle=\hat{a}^{{\dagger}}_{\rm CW}\hat{a}_{\rm
CW}+\hat{a}^{{\dagger}}_{\rm CCW}\hat{a}_{\rm CCW},$ (S25)
$\displaystyle\hat{\Theta}(\hat{b}^{{\dagger}}_{\rm CW}\hat{b}_{\rm
CW}+\hat{b}^{{\dagger}}_{\rm CCW}\hat{b}_{\rm CCW})\hat{\Theta}^{-1}$
$\displaystyle=\hat{b}^{{\dagger}}_{\rm CW}\hat{b}_{\rm
CW}+\hat{b}^{{\dagger}}_{\rm CCW}\hat{b}_{\rm CCW},$ (S26)
$\displaystyle\hat{\Theta}(\hat{p}^{2}+\hat{q}^{2})\hat{\Theta}^{-1}=\hat{p}^{2}\,\;\;\\!\\!\\!+$
$\displaystyle\,\;\;\\!\\!\\!\hat{q}^{2},\quad{\rm
for}\;\nabla\vartheta(\hat{q})=0.$ (S27)

However, this is not the case for interaction terms between optical modes.
According to Eqs. (S22) and (S23), we get the following transformation
relations

$\displaystyle\hat{\Theta}(\hat{a}^{{\dagger}}_{\rm CW}\hat{b}_{\rm
CCW}+\hat{a}_{\rm CW}\hat{b}_{\rm CCW}^{{\dagger}})\hat{\Theta}^{-1}$
$\displaystyle=e^{i(\theta_{4}-\theta_{1})}\hat{a}^{{\dagger}}_{\rm
CW}\hat{b}_{\rm CCW}+e^{i(\theta_{1}-\theta_{4})}\hat{a}_{\rm CW}\hat{b}_{\rm
CCW}^{{\dagger}},$ (S28) $\displaystyle\hat{\Theta}(\hat{a}^{{\dagger}}_{\rm
CCW}\hat{b}_{\rm CW}+\hat{a}_{\rm CCW}\hat{b}_{\rm
CW}^{{\dagger}})\hat{\Theta}^{-1}$
$\displaystyle=e^{i(\theta_{3}-\theta_{2})}\hat{a}^{{\dagger}}_{\rm
CCW}\hat{b}_{\rm CW}+e^{i(\theta_{2}-\theta_{3})}\hat{a}_{\rm CCW}\hat{b}_{\rm
CW}^{{\dagger}},$ (S29) $\displaystyle\hat{\Theta}(\hat{a}^{{\dagger}}_{\rm
CW}\hat{a}_{\rm CCW}+\hat{a}_{\rm CW}\hat{a}_{\rm
CCW}^{{\dagger}})\hat{\Theta}^{-1}$
$\displaystyle=e^{i(\theta_{2}-\theta_{1})}\hat{a}^{{\dagger}}_{\rm
CW}\hat{a}_{\rm CCW}+e^{i(\theta_{1}-\theta_{2})}\hat{a}_{\rm CW}\hat{a}_{\rm
CCW}^{{\dagger}},$ (S30)
$\displaystyle\hat{\Theta}(e^{i\varphi}\hat{b}^{{\dagger}}_{\rm
CW}\hat{b}_{\rm CCW}+e^{-i\varphi}\hat{b}_{\rm CW}\hat{b}_{\rm
CCW}^{{\dagger}})\hat{\Theta}^{-1}$
$\displaystyle=e^{i(\theta_{4}-\theta_{3}-\varphi)}\hat{b}^{{\dagger}}_{\rm
CW}\hat{b}_{\rm CCW}+e^{i(\theta_{3}-\theta_{4}+\varphi)}\hat{b}_{\rm
CW}\hat{b}_{\rm CCW}^{{\dagger}}.$ (S31)

Therefore, the gauge phases that make the Hamiltonian satisfy time-reversal
transformation relation must meet

$\displaystyle\theta_{4}-\theta_{1}=2\;\;\\!\\!\\!\pi\;\;\\!\\!\\!\mathbb{Z}_{1},\quad\theta_{3}-\theta_{2}=2\;\;\\!\\!\\!\pi\;\;\\!\\!\\!\mathbb{Z}_{2},\quad\theta_{2}-\theta_{1}=2\;\;\\!\\!\\!\pi\;\;\\!\\!\\!\mathbb{Z}_{3},\quad\theta_{3}-\theta_{4}+2\;\;\\!\\!\\!\varphi=2\;\;\\!\\!\\!\pi\;\;\\!\\!\\!\mathbb{Z}_{4},$
(S32)

where $\\{\mathbb{Z}_{1},\mathbb{Z}_{2},\mathbb{Z}_{3},\mathbb{Z}_{4}\\}$
represent arbitrary integers. Further processing Eq. (S32), we obtain the
necessary and sufficient condition for the system to hold time-reversal
symmetry, that is, $\varphi=\pi\;\;\\!\\!\\!\mathbb{Z}$ ($\mathbb{Z}$ denotes
arbitrary integer). This implies that the extra hopping phase $\varphi$
acquired by photons during tips-induced backscattering processes can also be
applied as a means to regulate the dynamics. For nontrivial $\varphi$, i.e.,
$\varphi\neq\pi\;\;\\!\\!\\!\mathbb{Z}$, as shown in Eqs. (S28)–(S31), its
value is opposite for the two reverse hopping routes, which is the root cause
of chirality.

###  II.3 Steady-state light intensity distributions

Although the analytical expression for chiral chaos is unavailable, we can
analytically obtain the steady-state light intensity distributions, which also
reveal the origin of chirality. For steady-state, Eqs. (S2)–(S6) can be
rewritten as

$\displaystyle 0=$ $\displaystyle-i(\Delta_{\rm A}-Gq_{s}-i\kappa)a_{\rm
CW}^{s}+i\eta a_{\rm CCW}^{s}+iJb_{\rm CCW}^{s}+\varepsilon_{1},$ (S33)
$\displaystyle 0=$ $\displaystyle-i(\Delta_{\rm A}-Gq_{s}-i\kappa)a_{\rm
CCW}^{s}+i\eta a_{\rm CW}^{s}+iJb_{\rm CW}^{s}+\varepsilon_{2},$ (S34)
$\displaystyle 0=$ $\displaystyle-i(\Delta_{\rm B}-i\gamma)b_{\rm
CW}^{s}+i|\xi|e^{i\varphi}b_{\rm CCW}^{s}+iJa_{\rm CCW}^{s},$ (S35)
$\displaystyle 0=$ $\displaystyle-i(\Delta_{\rm B}-i\gamma)b_{\rm
CCW}^{s}+i|\xi|e^{-i\varphi}b_{\rm CW}^{s}+iJa_{\rm CW}^{s},$ (S36)
$\displaystyle 0=$ $\displaystyle\ \Omega\>\\!\>\\!p_{s},\quad 0=-\
\\!\;\;\\!\\!\\!\Omega\>\\!q_{s}+G(|a_{\rm CW}^{s}|^{2}+|a_{\rm
CCW}^{s}|^{2})-\Gamma\>\\!p_{s}.$ (S37)

Solving the above equations, we get the following results

$\displaystyle a_{\rm CW}^{s}$
$\displaystyle=-i\dfrac{\varepsilon_{2}(\eta+Fe^{-i\varphi})+\varepsilon_{1}\widetilde{\Delta}_{\rm
A}}{\widetilde{\Delta}_{\rm
A}^{2}-(\eta+Fe^{-i\varphi})(\eta+Fe^{i\varphi})},\quad a_{\rm
CCW}^{s}=-i\dfrac{\varepsilon_{1}(\eta+Fe^{i\varphi})+\varepsilon_{2}\widetilde{\Delta}_{\rm
A}}{\widetilde{\Delta}_{\rm
A}^{2}-(\eta+Fe^{-i\varphi})(\eta+Fe^{i\varphi})},$ (S38) $\displaystyle
b_{\rm CW}^{s}$ $\displaystyle=\frac{J|\xi|e^{i\varphi}a_{\rm
CW}^{s}+J(\Delta_{\rm B}-i\gamma)a_{\rm CCW}^{s}}{(\Delta_{\rm
B}-i\gamma)^{2}-|\xi|^{2}},\quad b_{\rm
CCW}^{s}=\frac{J|\xi|e^{-i\varphi}a_{\rm CCW}^{s}+J(\Delta_{\rm
B}-i\gamma)a_{\rm CW}^{s}}{(\Delta_{\rm B}-i\gamma)^{2}-|\xi|^{2}},$ (S39)
$\displaystyle\qquad\qquad\qquad\qquad\quad p_{s}=\;\\!0,\quad
q_{s}=\dfrac{G}{\Omega}(|a_{\rm CW}^{s}|^{2}+|a_{\rm CCW}^{s}|^{2}),$ (S40)

in which

$\displaystyle\widetilde{\Delta}_{\rm A}=\Delta_{\rm
A}-\dfrac{G^{2}}{\Omega}(|a_{\rm CW}^{s}|^{2}+|a_{\rm
CCW}^{s}|^{2})-i\kappa-\dfrac{F(\Delta_{\rm B}-i\gamma)}{|\xi|},\quad
F=\dfrac{J^{2}|\xi|}{(\Delta_{\rm B}-i\gamma)^{2}-|\xi|^{2}}.$ (S41)

Therefore, the steady-state light intensities in resonator A $(I_{\rm
A}^{s}=|a_{\rm CW}^{s}|^{2}+|a_{\rm CCW}^{s}|^{2})$ corresponding to pumping
from port 1 and port 2 can be expressed as

$\displaystyle I_{\rm A,1}^{s}=g(|\widetilde{\Delta}_{\rm
A}|^{2}+|\eta+Fe^{i\varphi}|^{2}),\quad{\rm\
\;\;\\!\\!\\!\;\;\\!\\!\\!\;\;\\!\\!\\!\;\;\\!\\!\\!\;\;\\!\\!\\!pumping\
from\ port\ 1},$ (S42) $\displaystyle I_{\rm
A,2}^{s}=g(|\widetilde{\Delta}_{\rm
A}|^{2}+|\eta+Fe^{-i\varphi}|^{2}),\quad{\rm pumping\ from\ port\ 2}.$ (S43)

Here, $g=\varepsilon^{2}/|\widetilde{\Delta}_{\rm
A}^{2}-(\eta+Fe^{-i\varphi})(\eta+Fe^{i\varphi})|^{2}$ and we have omitted
$G^{2}(|a_{\rm CW}^{s}|^{2}+|a_{\rm CCW}^{s}|^{2})/\Omega$ (since
$G\ll\Omega)$. Similarly, the steady-state light intensities in resonator B
$(I_{\rm B}^{s}=|b_{\rm CW}^{s}|^{2}+|b_{\rm CCW}^{s}|^{2})$ can also be
obtained, but the specific forms are too cumbersome to list here. For the
convenience of comparison, we further define $\delta I_{\rm A}^{s}=I_{\rm
A,1}^{s}-I_{\rm A,2}^{s}$ as the steady- state light intensity difference in
resonator A caused by pumping from different ports. Obviously, when time-
reversal symmetry holds, i.e., $\varphi=\pi\;\;\\!\\!\\!\mathbb{Z}$, $\delta
I_{\rm A}^{s}=0$, and when time-reversal symmetry breaks, $\delta I_{\rm
A}^{s}=g\;\;\\!\\!\\!(|\eta+Fe^{i\varphi}|^{2}-|\eta+Fe^{-i\varphi}|^{2})$.
One can easily verify that $I_{\rm A,1}^{s}\Rightarrow$ $I_{\rm A,2}^{s}$ and
$\delta I_{\rm A}^{s}\Rightarrow-\;\;\\!\\!\\!\delta I_{\rm A}^{s}$ under the
transformation $\varphi\Rightarrow-\;\;\\!\\!\\!\varphi$. Such a
transformation relationship conforms the chiral symmetry of system dynamics on
the $\varphi\;\;\\!\\!\\!$-axis. Finally, it is also worth mentioning that the
steady-state solutions predict several other useful operating modes. For
example, scattering induced by intrinsic defects can be greatly suppressed by
properly selecting parameters, which is essential for constructing robust
channels. As shown in Eq. (S38), if $\eta+Fe^{i\varphi}\rightarrow 0$, then
the CCW mode in resonator A will be completely suppressed when pumping from
port 1. The same phenomenon also exists for the CW mode. Detailed discussion
on this aspect is beyond the scope of this Letter and will be undertaken in
the future.

###  II.4 Explanations of the structural parameters defined in the main text

Figure S4: Geometric interpretations for the structural parameters $S$ and
$C$. Here, the black dash-dotted lines represent $\lambda_{\rm max}=0$, and
the green and cyan regions separated by it represent $\lambda_{\rm max}>0$
(chaotic) and $\lambda_{\rm max}<0$ (ordered), respectively. In all panels,
the red and blue curves represent the maximal Lyapunov exponent spectra for
the corresponding extreme case (S, C reach 0 or 1), where in panels (b) and
(d), two different types of blue curves are presented as (1) and (2), since
they both meet the requirements for complete asymmetry or achirality. The
different positions of hands at the bottoms are indicators for the structural
features of the corresponding S or C.

We now provide detailed explanations and descriptions for the structural
parameters $S$ and $C$ defined in the main text. First, we briefly review the
Lyapunov exponents. (i) Lyapunov exponents describe the degree of separation
over time between neighboring trajectories in phase space. (ii) They have been
widely accepted as metrics to characterize the dynamics of nonlinear systems,
and many reliable calculation algorithms have been proposed S (13). (iii) In
general, their values are extremely sensitive to changes in parameters of the
differential or map equations. (iv) The sign of the maximal Lyapunov exponent
$\lambda_{\rm max}$ is opposite for ordered $(\lambda_{\rm max}<0)$ and
chaotic $(\lambda_{\rm max}>0)$ regimes, which ensures its excellent
distinguishing ability S (10). Second, we clarify the underlying assumptions
implicit in the structural parameters. (I) If the evolution of system dynamics
(order-chaos transitions, bifurcations, catastrophes, etc.) exhibits exactly
the same properties under two different sets of conditions, then the two sets
of conditions are considered to be equivalent and the corresponding evolution
processes are ideally symmetrical. (II) If the evolution of system dynamics
under two different sets of conditions are mirror images of each other, then
the two sets of conditions are considered as enantiomers and the corresponding
evolution processes are ideally chiral S (14).

Combining the Lyapunov exponents and the above-mentioned basic assumptions, we
propose the following metrics to measure the symmetry and chirality of any two
routes to chaos. Define $x$ as the condition under which the system is in
state $U$ with a maximal Lyapunov exponent $\lambda_{\rm max}$. Assign $N$
such conditions into set $\\{x_{i}\\}$ according to certain rules, we can
obtain an evolution process of system dynamics $\\{U_{i}\\}$ and the
corresponding maximal Lyapunov exponent array $\\{\lambda_{\rm max}^{i}\\},$
$i=1,2,\cdots\;\;\\!\\!\\!\\!,N$. Similarly, we create another $N$-element
condition set $\\{y_{i}\\}$, and obtain another evolution process
$\\{V_{i}\\}$ and a new maximal Lyapunov exponent array $\\{\Lambda^{i}_{\rm
max}\\},$ $i=1,2,\cdots\;\;\\!\\!\\!\\!,N$. Then, we define the following
structural parameters

$\displaystyle S=1-\dfrac{1}{N}\sum_{i=1}^{N}\dfrac{|\lambda^{i}_{\rm
max}-\Lambda^{i}_{\rm max}|}{|\lambda^{i}_{\rm max}|+|\Lambda^{i}_{\rm
max}|},\quad C=1-\dfrac{1}{N}\sum_{i=1}^{N}\dfrac{|\lambda^{i}_{\rm
max}-\Lambda^{N-i+1}_{\rm max}|}{|\lambda^{i}_{\rm max}|+|\Lambda^{N-i+1}_{\rm
max}|}.$ (S44)

Obviously, both parameters take values between $0$ and $1$ (inclusive).
According to the previous assumptions, if two evolution processes are ideally
symmetrical, i.e., $U_{i}=V_{i}$ and $\lambda^{i}_{\rm max}=\Lambda^{i}_{\rm
max}$, then $S=1$ [see Fig. S4(a)]. On the contrary, if two processes are
completely asymmetrical, that is, the operating states represented by each
pair of $U_{i}$ and $V_{i}$ are completely different (one is ordered and the
other is chaotic), then the signs of $\lambda^{i}_{\rm max}$ and
$\Lambda^{i}_{\rm max}$ must be opposite, resulting in $S=0$ [see Fig. S4(b)].
Note that the conditions for $S=1$ are stricter than those for $S=0$. As shown
in Figs. S4(a) and S4(b), for the process described by the red curve, there is
only one ideal symmetrical process, while there are countless completely
asymmetrical processes, e.g., (1) and (2) in Fig. S4(b). For the general case,
$S\in(0,1)$, which can be understood as the average degree of symmetry between
two evolution processes. In the same way, if two processes are ideally chiral,
i.e., $U_{i}=V_{N-i+1}$ and $\lambda^{i}_{\rm max}=\Lambda^{N-i+1}_{\rm max}$,
then $C=1$ [see Fig. S4(c)]. For completely achiral, one of $U_{i}$ and
$V_{N-i+1}$ is ordered and the other is chaotic, which means the signs of
$\lambda^{i}_{\rm max}$ and $\Lambda^{N-i+1}_{\rm max}$ are opposite,
resulting in $C=0$ [see Fig. S4(d)]. The conditions to be satisfied for $C=1$
are also stricter than those for $C=0$. As shown in Figs. S4(c) and S4(d), for
the process described by the red curve, there is only one ideal chiral
process, while there are countless completely achiral processes, e.g., (1) and
(2) in Fig. S4(d). For other cases, $C\in(0,1)$, which can be understood as
the average degree of chirality between two evolution processes. In
conclusion, the proposed structural parameters can properly quantify the
symmetry and chirality between routes to different chaos S (15).

##  III Details of chiral chaos for sensitive sensing

The core principle of the sensing method we proposed, i.e., constructing the
sensing window, has been explained in the main text. Below, we take the
detection of monochromatic optical (weak) signal as an example to test the
actual performance of sensing windows. More complicated cases, such as signals
with a certain bandwidth or signals with background noise, will be discussed
in our follow-up work. The specific form of the weak signal is

$s(\tau)=\;\;\\!\\!\\!\delta\varepsilon\;\;\\!\\!\\!{\rm
cos}\;\;\\!\\!\\![(\omega+\delta\omega)\tau+\theta\;\;\\!\\!\\!],\quad\delta\varepsilon=\sqrt{\dfrac{\kappa_{0}\;\;\\!\\!\\!p_{s}}{\hbar(\omega+\delta\omega)}}\;\;\\!\\!\\!,$
(S45)

where $\delta\varepsilon$, $\omega+\delta\omega$, and $p_{s}$ represent the
amplitude, frequency, and power of the weak signal, respectively. Here,
$\omega$ and $\varepsilon$ $(\varepsilon\gg\delta\varepsilon)$ denote the
frequency and amplitude of the pumping field, $\theta$ and $\delta\omega$
represent the initial phase difference and detuning between the signal and the
pumping field, respectively, $\kappa_{0}$ is the taper-resonator coupling
rate, and $\tau$ ($\tau=\Omega\;\;\\!\\!\\!t$) is the extended time scale. To
simplify subsequent discussions and facilitate experimental verification, we
assume that the weak signal is directly loaded into the pumping field. Then,
the equivalent Hamiltonian is described as

$\hat{H}=\left\\{\begin{split}&i\hbar\\!\\!\>\left[\varepsilon+\delta\varepsilon\;\;\\!\\!\\!e^{-i(\delta\omega\tau\;\\!+\;\\!\theta)}\;\\!\\!\right]\\!\\!\>\hat{a}_{\rm
CW}^{{\dagger}}+{\rm H.c.},\ {\rm input\ from\ port\ 1},\\\\[3.0pt]
&i\hbar\\!\\!\>\left[\varepsilon+\delta\varepsilon\;\;\\!\\!\\!e^{-i(\delta\omega\tau\;\\!+\;\\!\theta)}\;\\!\\!\right]\\!\\!\>\hat{a}_{\rm
CCW}^{{\dagger}}+{\rm H.c.},\ {\rm input\ from\ port\ 2},\end{split}\right.$
(S46)

where a rotating frame at frequency $\omega$ and the rotating-wave
approximation have been applied.

For a more intuitive comparison, we stimulate a conventional chaos-based
sensor S (16) (based on order-chaos phase transition) using the sensing
results when inputting from port 2 of our device. Actually, switching the
input port has no effect on the sensing performance of conventional reciprocal
chaos-based sensors. However, the situation is completely different in our
chiral chaotic sensor. After constructing a sensing window, we need to
superimpose the contributions from the two input ports to achieve the final
sensing result.

Figure S5: Panels (a) and (b) are phase diagrams of the system within the
selected parameter regions when pumping from port 1 and port 2, respectively.
Here, the green, red, and black regions represent the system locates in the
self-induced oscillations, period-doubling bifurcations, and chaotic phases,
respectively. The cyan region represents a special type of chaotic phase,
where the system is always in the chaotic phase no matter which port is
pumped. The parameters are $\Delta_{\rm A}=\Delta_{\rm B}=\Delta$,
$\delta\varepsilon=0$, $\varphi/\pi=$ $0.5$, $|\xi|/\Omega=5$, and other
parameters are the same as in Fig. S2.

Before testing, we need to construct the required sensing window. As shown in
the phase diagrams in Fig. S5, the operating states of the device are
different when pumping from different ports. It should be noted that a special
type of region exists on the phase diagrams, that is, the sensing window
region. In such a region, the system dynamics is sensitive to detected
signals, and this sensitivity is not affected by the randomness in certain
parameters of the signals. For example, the cyan region on the phase diagrams
is a sensing window region. In this region, the system is always in the
chaotic state no matter which port is pumped. Furthermore, for a resonant
($\delta\omega=0$) or near-resonant ($\delta\omega/\omega\ll 1$) signal with
amplitude $|\delta\varepsilon|$, even if the value of $\theta$ is random, the
system still can undergo a chaos-to-order phase transition upon pumping from
one of the ports. For $\theta\in(0,\pi/2)$, the chaos-to-order phase
transition can be induced when input from port 2, for $\theta\in(\pi/2,\pi)$,
the chaos-to-order phase transition can be induced when input from port 1. As
illustrated by the bidirectional arrows in Fig. S5, each vertical parameter
bar in the cyan region can be designed as a sensing window, and the half-width
and center position of the window are determined by the upper and lower
boundaries. It is obvious that the characteristics of sensing windows can be
adjusted smoothly over a wide range. Therefore, the application scope of such
sensing window region can be greatly broadened. Narrow windows are suitable
for sensitive sensing, while wide windows are suitable for threshold alarms S
(17). In our main text, the sensing window region constructed on the
$(\varepsilon,\omega)$ plane is located in the cyan region in Fig. S5, and the
operating point $F$ is at $\varepsilon/\Omega=54887$ and
$\Delta/\Omega=0.5598$.

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# Compressing Deep CNNs using Basis Representation and Spectral Fine-tuning

###### Abstract

We propose an efficient and straightforward method for compressing deep
convolutional neural networks (CNNs) that uses basis filters to represent the
convolutional layers, and optimizes the performance of the compressed network
directly in the basis space. Specifically, any spatial convolution layer of
the CNN can be replaced by two successive convolution layers: the first is a
set of three-dimensional orthonormal basis filters, followed by a layer of
one-dimensional filters that represents the original spatial filters in the
basis space. We jointly fine-tune both the basis and the filter representation
to directly mitigate any performance loss due to the truncation. Generality of
the proposed approach is demonstrated by applying it to several well known
deep CNN architectures and data sets for image classification and object
detection. We also present the execution time and power usage at different
compression levels on the Xavier Jetson AGX processor.

Index Terms—  Basis representation, network compression, orthogonal filters

##  1 Introduction

While there has been a tremendous surge in convolutional neural networks and
their applications in computer vision, relatively little is still understood
about how information is learned and stored in the network. This is evidenced
by the fact that researchers have successfully proposed different approaches
for compressing a network after it has been trained, including techniques like
pruning weights [1, 2], assuming row-column separability and applying low rank
approximations for computational gains [3, 4] and using basis representation
[5, 6, 7, 8, 4] to approximate the filter kernels. Although the results are
impressive, existing compression techniques are not always easy to implement
using standard deep learning tool boxes. In this paper, we are motivated by
the observation that the original filters can be represented as weighted
linear combinations of a set of 3D basis filters with one-dimensional weight
kernels as shown in Figure 1. While compression is achieved by using fewer
basis filters, we show that these basis filters can be jointly finetuned along
with the weight kernels to compensate for any loss in performance due to
truncation, and to thereby achieve state of the art results.

Fig. 1: The network is compressed by replacing each convolution layer (top) in
a CNN with the basis convolution layer (bottom). Q basis filters are
initialized with the top ranked eigen vectors of P original filters, with Q
$<$ P.

The representation of spatial filters as a linear combination of orthogonal
basis filters is also known as spectral decomposition, and the weights
comprise the corresponding spectra of the filters. Since our approach updates
the basis filters and their weighted contribution to the overall result, we
refer to this process as Spectral Fine Tuning (SFT). Not only does SFT reduce
the number of learnable parameters, but also the weights are statistically
uncorrelated, and therefore adapt much faster than the conventional finetuning
of spatial domain filters. Figure 2 shows the main steps in our proposed
approach for network compression.

Fig. 2: Schematic diagram of our compression framework.

Low rank decomposition has been extensively used for neural network
compression in the past, but our work is differs from previously published
papers in the manner in which the filters are represented and finetuned using
basis filters and weights. For example methods such as [6] enforce filter rank
reduction while training the network, thereby making it more compressible than
its normally trained counterpart. In contrast, [4] achieves compression by
combining channel wise low rank approximation and with separable one-
dimensional spatial filters. In [7] multiply-accumulate operations are reduced
via constrained optimization. Low rank factorization and pruning are combined
in [8] by cascading the low rank projections of filters in the current layer
to the next layer. In comparison, our method is relatively straight forward
(as depicted in Figure 2), and yet provides competitive results.

##  2 Basis Representation and Learning

Consider the fundamental convolution operation in any given layer of a
convolutional neural network. Assume that an input block of data $x(m,n,l)$
(such as the activations or output of the previous layer) is convolved with a
set of 3D filters $h_{k}(m,n,l),\ k=1$…$P$. The output $y_{k}(m,n)$ can be
expressed as $y_{k}(m,n)=x(m,n,l)\ *\ h_{k}(m,n,l),\qquad 1\leq k\leq p$ where
$*$ represents the convolution operation. The filters can be expressed as a
linear combination of Q basis functions $f_{i}(m,n,l),\quad i=1$…$Q$, such
that $h_{k}(m,n,l)=\sum_{i=1}^{Q}w_{k}(i)\cdot f_{i}(m,n,l)$ where $w_{k}(i)$
are the weights of linear combination. Using this representation, the output
can be expressed as

$y^{\prime}_{k}(m,n)=\sum_{i=1}^{Q}w_{k}(i)\cdot[x(m,n,l)*f_{i}(m,n,l)],\quad
1\leq k\leq P$ (1)

The key observation is that the Q convolution terms
$z_{i}(m,n)=x(m,n,l)*f_{i}(m,n,l)$ need to be computed only once, and they are
common to all $P$ outputs $y^{\prime}_{k}(m,n)$. These can be stacked together
to form the 3D intermediate result $z(m,n,i)$ while the weights can be treated
as $1\times 1\times Q$ filter $w_{k}(i)$. Therefore, the outputs
$y^{\prime}_{k}(m,n)$ are simply the convolution of two, i.e

$y^{\prime}_{k}(m,n)=w_{k}(i)*z(m,n,i)$ (2)

We refer to this construct using two successive convolutions as BasisConv.

Compression of pretrained Networks and Spectral Fine Tuning: SFT is the
simultaneous learning of $w_{k}(i)$ and $f_{i}(m,n,l)$ in Equation 1 (across
all layers of the network) to mitigate overall performance loss due the choice
of $Q<P$. We now discuss how the basis filters $f_{i}(m,n,l)$ and spectral
weights $w_{k}(i)$ are initialized, and the orthogonality criteria introduced
in the loss function to ensure that the basis representation condition is
preserved.

It is well known that eigen decomposition results in a compact basis that
minimizes the reconstruction error achieved by a linear combination of basis
functions. We therefore initially choose $f_{i}(m,n,l)$ as the eigen filters
that represent the sub-space in which the original filters $h_{k}(m,n,l)$ lie.
The method for obtaining these is also well-known and straightforward.
Basically, we define the $LD^{2}\times 1$ dimensional vector $\mathbf{h_{k}}$
as a vectorized representation of $h_{k}(m,n,l)$, and construct the matrix
$\mathbf{A}=[\mathbf{h_{1}}\ \mathbf{h_{2}}\ ...\ \mathbf{h_{P}}]$ with
$\mathbf{h_{k}}$ as its columns. The eigenvectors of $\mathbf{AA^{T}}$
represent the sub-space of the filters, and satisfy the relation
$\mathbf{AA^{T}f_{i}}=\lambda_{i}\mathbf{f_{i}}$, where $\mathbf{f_{i}}$ are
the eigenvectors, and $\lambda_{i}$ are the corresponding eigenvalues. The
eigen filter $f_{i}(m,n,l)$ is readily obtained by re-ordering the elements of
the eigenvector $\mathbf{f_{i}}$ into a $D\times D\times L$ array. It should
be noted that these matrix-vector manipulations do not alter the inherent
tensor structure and relations between the individual elements of the 3D
filters. We select a small subset of eigenvectors which correspond to the
largest Q eigen-values that best represent the dominant coordinates of the
filters’ subspace using the metric
$t=\frac{\sum_{i=1}^{Q}\lambda_{i}}{\sum_{i=1}^{LD^{2}}\lambda_{i}}$ to choose
Q such that most of the relevant information is retained in the selected
eigen-vectors. Using this criteria, $Q$ is chosen for each layer such that it
results in no more than 3% drop in overall performance.

The decomposition of the filters $h_{k}$ of any given layer of the network can
be succinctly expressed in matrix vector notation by defining
$\mathbf{F}=[\mathbf{f_{1}}\ \mathbf{f_{2}}\ ...\ \mathbf{f_{Q}}]$ (i.e. the
matrix of eigenvectors of the filters for that layer) so that
$\mathbf{h_{k}}=\mathbf{Fw_{k}}$ and $\mathbf{w_{k}}=[w_{k}(1)\ w_{k}(2)\ ...\
w_{k}(Q)]^{T}$ is a $Q\times 1$ vector of weights. Since $\mathbf{F^{T}F=I}$
(i.e. the identity matrix), the weights are easily obtained by computing
$\mathbf{w_{k}}=\mathbf{F^{T}h_{k}}$.

SFT converges much faster than conventional finetuning of the original spatial
filters. This is partly due to the fact that the elements of the weight
vectors $\mathbf{w_{k}}$ are statistically uncorrelated. This is easy to show
by noting that $\mathbf{A}=\mathbf{F}[\mathbf{w_{1}}\ \mathbf{w_{2}}\ ...\
\mathbf{w_{P}}]=\mathbf{FW}$ where $\mathbf{W}=[\mathbf{w_{1}}\
\mathbf{w_{2}}\ ...\ \mathbf{w_{P}}]$ is a matrix whose columns are the weight
vectors associated with the filters in the corresponding columns of
$\mathbf{A}$. Therefore, $\mathbf{AA^{T}}=\mathbf{FWW^{T}F^{T}}$. It is also
true that $\mathbf{AA^{T}}=\mathbf{F{\Delta}F^{T}}$ where $\mathbf{\Delta}$ is
a diagonal matrix with the eigenvalues $\lambda_{i}$ as its diagonal elements.
Hence, $\mathbf{WW}^{T}=\mathbf{\Delta}$ is a diagonal matrix and
$\sum_{k=1}^{P}{w_{k}(i)}^{2}=\lambda_{i},\text{ and
}\sum_{k=1}^{P}{w_{k}(i)}{w_{k}(j)}=0$. Therefore the convergences of each
element of $\mathbf{w_{k}}$ is statistically uncorrelated with the behavior of
the other elements, which implies that the corresponding eigen vector also
contributes to the overall learning process in an uncorrelated manner. Thus
far, we have discussed fine-tuning the weights while holding the basis filters
fixed as eigen-vectors of the original filters. However refining the basis
filters can improve performance of the compressed network. To finetune the
basis filters, we must ensure that they remain an orthonormal set (so that the
weights continue to be a spectral representation in the new basis). This is
achieved by including the following term in the overall cost function

$\mathbf{J}_{f}=\frac{\alpha}{{Q}}\sum_{i=1}^{Q}(1-{\mathbf{f}_{i}}^{T}\mathbf{f}_{i})^{2}+\frac{2(1-\alpha)}{Q(Q-1)}\sum_{i=1}^{Q}\sum_{j=i+1}^{Q}({\mathbf{f}_{i}}^{T}\mathbf{f}_{j})^{2}$
(3)

where $\mathbf{f}_{i}$ are the individual basis filters, $Q$ is the number of
such filters in a given layer, and $a$ is a positive number between 1 and 0.
The first term in $\mathbf{J}_{f}$ causes the $L2$ norm of
$\mathbf{f}_{i},i=1...Q$ to be as close to unity as possible, while the second
term ensures the filters remain orthogonal. The parameter $a$ can be set to
0.5 to equally emphasize both term, or selected as necessary to balance the
trade-off between the two terms. Finally, $f_{i}(m,n,l)$ and ${w_{k}}(i)$ are
both jointly fine-tuned to minimize the sum of $\mathbf{J}_{f}$ and the cost
function (such as cross-entropy loss) used for training the original network.

Method | 3$\times$ | 4$\times$ | 5$\times$
---|---|---|---
Jaderberg et. al.[5] | 87.6 | 80.2 | 60.2
Zhang et. al.[9] (asym.) | 89.5 | 89.0 | 87.9
Zhang et. al.[9] (asym. fine-tuned) | 89.9 | 89.6 | 88.9
CP (fine-tuned) [10] | - | 88.9 | 88.2
ELR [11] | - | 89.4 | 88.8
Li et. al. [12] | - | 81.3 | 67.9
Li et. al. [7] | 88.9 | - | -
Minnehan et. al. [8] | - | - | 89.4
Ours | 90.6 | 89.6 | 89.4
Table 1: Top-5 accuracy of VGG16 pretrained on ImageNet and compressed to 3x, 4x and 5x speedup ratios. Model | FLOPs (Billions) | Params (Millions) | # of Filters | Accuracy (%)
---|---|---|---|---
CNN | BasisNet |  $\downarrow$% | CNN | BasisNet |  $\downarrow$% | CNN | BasisNet |  $\downarrow$% | CNN | BasisNet | $\downarrow$
Alexnet | 0.03 | 0.01 | 76.20 | 2.47 | 0.68 | 72.33 | 1152 | 256 | 77.78 | 43.87 | 44.68 | -0.81
VGG16 | 0.63 | 0.12 | 80.44 | 14.71 | 1.67 | 88.68 | 4224 | 501 | 88.14 | 68.72 | 69.53 | -0.81
Resnet110 | 0.51 | 0.16 | 68.65 | 1.73 | 0.63 | 63.41 | 4144 | 1271 | 69.33 | 71.99 | 71.67 | 0.32
Densenet190 | 18.61 | 3.44 | 81.52 | 25.60 | 6.98 | 72.73 | 20290 | 4116 | 79.71 | 82.83 | 82.28 | 0.55
Table 2:  Results of compressing four different networks on CIFAR100 shows
significant FLOPs and parameter reduction is achieved with minimal loss of
accuracy. [$\downarrow$%] denotes percent decrease in the metric compared to
the baseline model.

##  3 Experiments

Method | FLOPs | Params. | Accuracy (%)
---|---|---|---
(%$\downarrow$) |  (%$\downarrow$) | Top1 | Top5
SFP [13] | 41.80 | - | 74.61 | 92.06
IMP [14] | 45.00 | 51.48 | 74.50 | -
CP [10] | 50.00 | - | - | 90.80
LFC [15] | 50.00 | - | 73.40 | 91.40
ELR [11] | 50.00 | - | - | 91.20
GDP [16] | 51.30 | - | 71.89 | 90.719
FPGM [17] | 53.50 | - | 74.83 | 92.32
DCP [18] | 55.76 | 51.45 | 74.95 | 92.32
GBN [19] | 55.06 | 53.40 | 75.18 | 92.41
LFPC [20] | 60.80 | - | 74.46 | 92.04
HRank [21] | 62.10 | 46.13 | 71.98 | 91.01
Alvarez et. al. [6] | - | 27.0 | 75.2 | 91.01
TRP1+Nu [4] | - | 44.0 | 74.06 | 92.07
Ours (A) | 51.28 | 49.28 | 75.85 | 92.84
Ours (B) | 62.68 | 61.16 | 74.96 | 92.44
Table 3: Results for Resnet-50 pretrained on ImageNet. Our compression configuration (A) achieves higest Top-1 and Top-5 accuracy at comparable level of compression. Method | FLOPs $\downarrow$% | Params $\downarrow$% | Acc. %
---|---|---|---
Li et. al. [12] | 27.6 | 13.7 | 93.06
NISP [22] | 43.6 | 42.6 | 93.09
DCP-A [18] | 47.1 | 70.3 | 93.10
CP [10] | 50.0 | - | 91.80
AMC [23] | 50.0 | - | 91.90
GBN [19] | 60.1 | 53.5 | 93.43
LFPC [20] | 52.9 | - | 93.24
FPGM [17] | 52.6 | - | 93.26
Minnehan et. al [8] | 48.8 | - | 93.22
TRP1 [4] | 56.7 | - | 92.77
Ours | 64.1 | 60.2 | 93.01
Table 4: Results for Resent-56 on CIFAR10. Our compressed model achieves
highest compression while attaining comparable or in some cases better
accuracy.

Data Sets and CNN Architectures: We performed our experiments on various
publicly available datasets. These include image classification datasets
CIFAR10, CIFAR100, ImageNet and object detection dataset MS-COCO.

VGG16 on ImageNet: We tested our method on VGG16 pretrained on ImageNet by
compressing the model to 3x, 4x and 5x speedup ratios. Results of this
experiment are presented in Table 1. Compared with other methods our basis
VGG16 achieves better Top-5 accuracy on all speedup ratios.

CIFAR-100 with Different Networks: Table 2 shows the results of compressing
four different networks using the proposed method, and their performance on
the CIFAR-100 data set. For instance, compressing DenseNet190 reduces FLOPS by
81%, and the number of parameters by 73% with a minor loss of 0.55% in
performance accuracy. Similar gains are also noted for the other architectures
as well.

Resnet50 on ImageNet: We also tested our method on Resnet with 50 convolution
layers, pretrained on ImageNet by compressing the model at two different
levels of compression: 51.28% and 61.68%. Results of this experiment and
comparisons with other methods are presented in Table 3. When compared with
existing state of art methods like IMP[14], CP[10], FPGM[17], GBN[19],
LFPC[20] and HRank[21] our model (A) achieves better accuracy for the similar
levels of compression. Our second compression configuration (B) achieves best
compression both in terms of FLOPs and trainable parameters, while keeping
Top-1 and Top-5 accuracy comparable to other methods.

Resnet56 on CIFAR10: For CIFAR10 dataset we tested our method on Resnet56
model and compared it with other similar methods. Comparative Results are
shown in Table 4.

Results on Object Detection: To demonstrate our method’s effectiveness
irrespective of the model architecture, dataset and the objective function, we
tested it on object detection problem. We used 2 predefined variants of YOLOv3
[24] object detector pretrained on MS-COCO dataset. These variants are
YOLOv3-tiny [24], which contains 13 convolution layers and a much deeper model
YOLOv3-416 [24], which contains 75 convolution layers. For YOLOv3-tiny our
method was able to achieve 51.26% reduction in FLOPs and 67.73% reduction in
model parameters with 29.6 MAP while the MAP for baseline model is 32.9. We
also compressed YOLO-416 to 2x and 3x speedup ratios. These compressed models
attain 53.83 and 55.34 MAP score respectively, as compared to the baseline
model which has 55.40 MAP.

Method | Baseline | 2$\times$ | 3$\times$ | 4$\times$ | 5$\times$
---|---|---|---|---|---
Inference time (ms) | 40 | 30 | 26 | 25 | 23
Power (W) | 6.01 | 5.55 | 5.15 | 5.08 | 4.94
Energy (Joule) | 0.24 | 0.17 | 0.14 | 0.12 | 0.11
Top5 Accuracy (%) | 90.4 | - | 90.6 | 89.6 | 89.4
Table 5: VGG16 inference time, power and energy consumption on NVIDIA Jetson
AGX Xavier, with various compression factors of 2x, 3x, 4x and 5x

Implementation On Edge Processors: We also measured the GPU speed up of our
method on Nvidia Jetson Nano by compressing the VGG16 to reduce the FLOPs by
50% and 66%. Our method achieves 35.6% and 45.3% speedup respectively. Table 5
shows the inference time, power and energy consumption of VGG16 on the NVIDIA
Jetson AGX Xavier (trained on ImageNet) using compression factors of 2x, 3x,
4x and 5x. We see that total energy use can be reduced in half with less than
1% loss in accuracy. Thus, we verified that the compressed model not only runs
faster, but also requires less energy when implemented in edge processors that
operate with limited resources such as GPU memory and power. It should be
noted that we did not implement our method in C++ but used the python based
Pytorch instead. We believe that an optimized C++ cuda implementation can
achieve even better performance on GPUs.

##  4 Summary and Discussions

We have presented a straightforward yet effective method for compressing deep
CNNs that outperforms most other state of the art techniques in terms of
reduction in FLOPS and number of parameters with negligible performance loss.
Despite the overwhelming interest in neural network compression in recent
years, there is still a lack of standardized definitions for metrics such as
FLOPs. Some methods choose to compress the network to arbitrary reduction in
FLOPs and report the model’s accuracy, while others compress the network in
such a way that accuracy of the resulting model is within a certain range of
base model. The inference time of the compressed model is an important metric
that should be considered, but introduces its own complexities due to
dependencies on libraries and hardware platform. We also note that power
utilization is an important consideration for edge processors, and should be
included as a metric for comparing compressed networks.

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