# Quantum synchronization via Active-Passive-Decomposition configuration: An
open quantum system study

Nan Yang<EMAIL_ADDRESS>Center for Quantum Science and Engineering, and
Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey
07030, USA  Ting Yu  Center for Quantum Science and Engineering, and
Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey
07030, USA

###### Abstract

In this paper, we study the synchronization of dissipative quantum harmonic
oscillators in the framework of quantum open system via the Active-Passive
Decomposition (APD) configuration. We show that two or more quantum systems
may be synchronized when the quantum systems of interest are embedded in
dissipative environments and influenced by a common classical system. Such a
classical system is typically termed as a controller, which (1) can drive
quantum systems to cross different regimes (e.g., from periodic to chaotic
motions) and (2) constructs the so-called Active-Passive Decomposition
configuration such that all the quantum objects under consideration may be
synchronized. The main findings of this paper is that we demonstrate that the
complete synchronizations measured by the standard quantum deviation may be
achieved for both stable regimes (quantum limit circles) and unstable regimes
(quantum chaotic motions). As an example, we numerically show in an
optomechanical setup that the complete synchronization can be realized in
quantum mechanical resonators.

##  I Introduction

The history of the synchronization can be traced back to the Dutch scientist,
C. Huygens, who firstly recorded the synchronization behaviors of two
suspended pendulums Hugenii (1673). He also accurately understood the physics
behind the phenomena: the oscillators may adjust their rhythms to reach a
consistent state due to weak interactions. Since then, the synchronization
problems have been recognized as a universal phenomenon in nature ranging from
bursting neurons, fireflies, and chemical reactions, to human activities,
seasonal migrations, and solar systems, and the references therein Pikovsky et
al. (2001); Strogatz (2004). In the domain of classical physics, for example,
the synchronization of periodic oscillators may be studies based on the Adler
equation Adler (1946) and the Kuramoto model Kuramoto (1975). In addition, the
chaotic synchronization has also been extensively studied by many different
methods Pecora and Carroll (1990); Kocarev and Parlitz (1995); Parlitz et al.
(1996); Rosenblum et al. (1996); Rosa et al. (1998); Rulkov et al. (1995);
Kocarev and Parlitz (1996); Rosenblum et al. (1997). Among them, the so-called
Active-Passive Decomposition (APD) configuration Kocarev and Parlitz (1995);
Parlitz et al. (1996) provides a general method for the complete
synchronization of chaotic systems.

The synchronization problems in quantum systems are also of great interests
due to their practical applications and close connections with the fundamental
aspects of quantum physics such as the transition from quantum to classical
domains. The related works have been reported in various quantum systems
including atomic systems Xu et al. (2014); Karpat et al. (2020); Siwiak-Jaszek
et al. (2020); Jaseem et al. (2020), qubits Zhirov and Shepelyansky (2008,
2009); Giorgi et al. (2016); Cattaneo et al. (2021), cold irons Hush et al.
(2015), spin models Orth et al. (2010); Bellomo et al. (2017); Roulet and
Bruder (2018); Giorgi et al. (2013); Karpat et al. (2019); Tindall et al.
(2020); Goychuk et al. (2006), lattices and dimer Cabot et al. (2019);
Michailidis et al. (2020); Siwiak-Jaszek and Olaya-Castro (2019), Van der Pol
oscillators Lee and Sadeghpour (2013); Lee et al. (2014); Walter et al.
(2015); Sonar et al. (2018); Eneriz et al. (2019); Mok et al. (2020); Yuzuru
and Nakao (2021), quantum harmonic oscillators Giorgi et al. (2012); Manzano
et al. (2013); Benedetti et al. (2016); Davis-Tilley and Armour (2016); Walter
et al. (2014); Makino et al. (2016); Lörch et al. (2016a); Lörch et al.
(2017); Nigg (2018); Qiao et al. (2018); Wächtler et al. (2020); Koppenhöfer
et al. (2020), and optomechanical systemsAmitai et al. (2017); Liao et al.
(2019); Heinrich et al. (2011); Ludwig and Marquardt (2013); Weiss et al.
(2016); Li et al. (2016). Quantum synchronization in non-Markovian
environments were studied in Ref. Karpat et al. (2021). It has been
experimentally demonstrated that high frequency resonators may potentially
enable the quantum synchronization Zhang et al. (2012); Bagheri et al. (2013);
Matheny et al. (2014); Zhang et al. (2015); Lörch et al. (2016b); Shlomi et
al. (2015); Gil-Santos et al. (2017). More applications of quantum
synchronization in quantum network and key-distribution may be found in
Kwasigroch and Cooper (2017); Pljonkin et al. (2017). Despite remarkable
progress, many interesting problems involving quantum open systems still need
to be carefully studied. Particularly, how to synchronize quantum systems in
unstable regimes is a long-standing problem.

Figure 1: (color online) Criterion for quantum synchronization of two systems
$1$ and $2$ is characterized by the identity of their standard deviations
($\sigma_{1,x},\sigma_{1,p}$) and ($\sigma_{2,x},\sigma_{2,p}$).

The purpose of this paper is studying the synchronization of two quantum
harmonic oscillators in the quantum open system framework. We demonstrate the
quantum synchronization may be achieved for the dissipative quantum systems in
not only stable regimes, but also the more complex unstable regimes where the
chaotic dynamics is involved.

For our quantum harmonic oscillator model, it is convenient to use the
standard deviations $\sigma_{x}$, $\sigma_{p}$ of the position and momentum
operators $\hat{x}$, $\hat{p}$ to measure the degree of quantum
synchronization. As shown in this paper, the given quantum synchronization
criterion allows us to analytically demonstrate the synchronization between
two quantum harmonic oscillators in two separate dissipative environments. Our
approach have used the Active-Passive Decomposition (APD) configuration
Kocarev and Parlitz (1995); Parlitz et al. (1996) to show that the
participating classical system can enable the quantum synchronization through
constructing an APD configuration and drives the quantum systems into a
desired type of motion (e.g., from period to chaos). For the continuous-
variable systems considered in this paper, we can reconstruct classical
trajectories from the quantum systems Takens (1981); Yang et al. . Finally, we
also numerically show in an experimentally accessible optomechanical setup
Aspelmeyer et al. (2014); Sciamanna (2016); Monifi et al. (2016); Bakemeier et
al. (2015); Buters et al. (2015); Carmon et al. (2007); Larson and Horsdal
(2011); Lee et al. (2009); Lü et al. (2015); Wang et al. (2016); Ma et al.
(2014); Marino and Marin (2013); Navarro-Urrios et al. (2017); Piazza and
Ritsch (2015); Sun and Sukhorukov (2014); Suzuki et al. (2015); Walter and
Marquardt (2016); Wang et al. (2014); Liao and Nori (2013); Wu et al. (2017);
Yang et al. (2015); Zhang et al. (2010, 2021); Liu et al. (2017); Wang et al.
(2019); Qin et al. (2021) that the desirable synchronization can be
accomplished .

This paper is organized as follows. In Sec. II, we introduce the criterion of
quantum synchronization for our continuous-variable systems. In Sec. III, we
discuss the realization of quantum synchronizations in several interesting
scenarios. Then, in Sec. IV, we apply our quantum synchronization approach to
an optomechanical setup; our numerical simulations show that the quantum
synchronization is achieved in two quantum mechanical resonators in both
periodic and chaotic regimes. Finally, we summarize and conclude this paper in
Sec. V.

##  II Criteria for quantum synchronization

In classical dynamics, the synchronization of two oscillators can be measured
by the relative relationships between their trajectories, e.g., complete
synchronization refers to approaching identical trajectories. Quantum systems
by default would not allow us to use concepts like ”classical trajectories”
directly Mari et al. (2013). In quantum domains, various definitions for
synchronization have been proposed, such as the semiclassical trajectory
approach Mari et al. (2013), the relative phase distribution Hush et al.
(2015) and Wigner function approach Makino et al. (2016). Also, it should be
noted that a definition based on observables has been provided recently for
the synchronization of quantum limit circles Buca et al. (2022).

For our discussions, the criteria to be used to measure the quantum
synchronization for continuous variables are based on the standard deviations
[$\sigma_{x}(t)$, $\sigma_{p}(t)$] of the position and momentum operators
($\hat{x}$, $\hat{p}$) of the harmonic oscillator systems:
$\sigma_{x}=\sqrt{\langle\hat{x}^{2}\rangle-\langle\hat{x}\rangle^{2}}$ and
$\sigma_{p}=\sqrt{\langle\hat{p}^{2}\rangle-\langle\hat{p}\rangle^{2}}$. Of
course if the quantum system under consideration has other degrees of freedom
such as spins or discrete systems, our definition of the synchronization must
be modified accordingly Xu et al. (2014); Karpat et al. (2020); Siwiak-Jaszek
et al. (2020); Jaseem et al. (2020); Zhirov and Shepelyansky (2008, 2009);
Giorgi et al. (2016); Cattaneo et al. (2021); Hush et al. (2015); Orth et al.
(2010); Bellomo et al. (2017); Roulet and Bruder (2018); Giorgi et al. (2013);
Karpat et al. (2019); Tindall et al. (2020); Goychuk et al. (2006). In this
way, the synchronization of two quantum systems may be measured by the
relative relationships between their standard deviations [$\sigma_{1,x}(t)$,
$\sigma_{1,p}(t)$] and [$\sigma_{2,x}(t)$, $\sigma_{2,p}(t)$] of the quantum
position and momentum ($\hat{x}$, $\hat{p}$).

To introduce the detailed definition for the quantum synchronization, we first
review the definition of complete synchronization in classical systems
Pikovsky et al. (2001); Pecora and Carroll (1990).

_Complete synchronization_ in classical contexts refers to the identity of the
trajectories of two dynamical systems. Consider two autonomous dynamical
systems $\dot{\textbf{y}}_{1}=\textbf{f}(\textbf{y}_{1})$ and
$\dot{\textbf{y}}_{2}=\textbf{f}(\textbf{y}_{2})$, where $\textbf{y}_{1}$ and
$\textbf{y}_{2}$ are $N$-dimensional variables governed by the function
$\textbf{f}:R^{N}\rightarrow R^{N}$. Here, $\textbf{y}_{1}$ and
$\textbf{y}_{2}$ are called complete synchronization if and only if their
difference
$\lim_{t\rightarrow\infty}\|\textbf{y}_{1}(t)-\textbf{y}_{2}(t)\|=0$.

In the similar spirit, here we show that the above definition can be directly
extended to quantum systems when the standard deviations are used.

(Complete synchronization) Two quantum systems $1$ and $2$ are called complete
synchronization if their standard deviations [$\sigma_{1,x}(t)$,
$\sigma_{1,p}(t)$] and [$\sigma_{2,x}(t)$, $\sigma_{2,p}(t)$] satisfy the
conditions:

$\lim_{t\rightarrow\infty}\|\sigma_{1,x}(t)-\sigma_{2,x}(t)\|=0,\ \ \
\lim_{t\rightarrow\infty}\|\sigma_{1,p}(t)-\sigma_{2,p}(t)\|=0,$ (1)

where [$\sigma_{1,x}(t)$, $\sigma_{1,p}(t)$] and [$\sigma_{2,x}(t)$,
$\sigma_{2,p}(t)$] represent the standard deviations of the two quantum
systems $1$ and $2$.

In the next section, we will use these criteria to discuss some physically
interesting examples that can display the realization of quantum
synchronizations.

##  III Synchronization of two quantum harmonic oscillators based on Active-
Passive Decomposition configuration

Figure 2: (color online) The schematic diagram for the synchronization of two
quantum harmonic oscillators $\hat{b}_{1}$ and $\hat{b}_{2}$.

To begin with, let us notice that, for classical dynamical systems, the so-
called complete synchronization may be accomplished through many different
ways such as mutual interactions, common driving forces, and feedback control
mechanisms, etc. Interestingly, the synchronization has been extended to some
important nonlinear systems where the chaotic synchronization may be observed.
Among several useful methods in realizing chaotic synchronization, we find the
Active-Passive Decomposition (APD) configuration particularly convenient for
our discussions.

To put our discussions into perspective, let us first briefly review the basic
idea of Active-Passive Decomposition (APD) configuration. In the APD model,
two chaotic subsystems to be synchronized can be written as non-autonomous
forms:

$\dot{\textbf{z}}_{1}=\textbf{f}[\textbf{z}_{1},\textbf{s}(t)],\qquad\dot{\textbf{z}}_{2}=\textbf{f}[\textbf{z}_{2},\textbf{s}(t)],$
(2)

where the dynamics of both $\textbf{z}_{1}$ and $\textbf{z}_{2}$ are ruled by
the function f, and $\textbf{s}(t)$ is the common external driving governed by
the autonomous function $\dot{\textbf{s}}(t)=\textbf{h}[\textbf{s}(t)]$. Here,
the synchronization differences e is defined as
$\textbf{e}=\textbf{z}_{1}-\textbf{z}_{2}$, and its dynamics is ruled by

$\dot{\textbf{e}}=\textbf{f}[\textbf{z}_{1},\textbf{s}(t)]-\textbf{f}[\textbf{z}_{1}-\textbf{e},\textbf{s}(t)].$
(3)

The APD model constructs a configuration that Eq. (3) is asymptotically stable
at the zero point $\textbf{e}=0$. Thus, the synchronization differences e goes
to zero as the time increases and complete synchronization occurs for two
chaotic dynamical systems $\textbf{z}_{1}$ and $\textbf{z}_{2}$.

As to be shown below, the Active-Passive Decomposition (APD) configuration
Kocarev and Parlitz (1995); Parlitz et al. (1996) may also be applied to the
synchronization of quantum systems. As shown in Fig. 2, our quantum model
consists of two separate quantum harmonic oscillators ($\hat{b}_{1}$ and
$\hat{b}_{2}$) in different heat baths and a classical controller that
produces a common force $s(t)$ acting on $\hat{b}_{1}$ and $\hat{b}_{2}$
simultaneously. Here, the effective Hamiltonian of the two harmonic
oscillators is given by

$H=\hbar[\Omega_{1}+g_{1}s(t)]\hat{b}^{\dagger}_{1}\hat{b}_{1}+\hbar[\Omega_{2}+g_{2}s(t)]\hat{b}^{\dagger}_{2}\hat{b}_{2},$
(4)

where $\Omega_{j}$ is the resonant frequency of the $j-th$ quantum harmonic
oscillator $\hat{b}_{j}$. The term $g_{j}s(t)$ refers to the frequency shift
of $\hat{b}_{j}$ brought by the classical input $s(t)$, and $g_{j}$ is the
coupling strength. In this setting, the quantum dynamics are modulated by the
classical controller, which can be driven from periodic to chaotic regimes by
adjusting the classical input $s(t)$. In what follows, we first derive the
equations of motions for the second order terms of the quantum harmonic
oscillators ($\hat{b}_{1}$ and $\hat{b}_{2}$), which determine the dynamics of
the corresponding standard deviations [$\sigma_{1,x}$, $\sigma_{1,p}$] and
[$\sigma_{2,x}$, $\sigma_{2,p}$]. Then, we will demonstrate that the quantum
synchronization of two harmonic oscillators is achievable and stable in the
APD configuration.

###  III.1 The equations of motions for the second order terms of the quantum
harmonic oscillators $\hat{b}_{1}$ and $\hat{b}_{2}$

For convenience, we use the shifted quantum harmonic oscillators $\hat{b}_{1}$
and $\hat{b}_{2}$

$\hat{b}_{1}=\beta_{1}+\hat{\tilde{b}}_{1},\ \ \
\hat{b}_{2}=\beta_{2}+\hat{\tilde{b}}_{2},$ (5)

where $\beta_{j}=\langle\hat{b}_{j}\rangle$ refers to the classical mean value
and $\hat{\tilde{b}}_{j}$ is the quantum fluctuation term of the mechanical
mode $\hat{b}_{j}$ ($\langle\hat{\tilde{b}}_{j}\rangle=0$). Then, the
Hamiltonian in terms of $\hat{\tilde{b}}_{j}$ $(j=1,2)$ is obtained by
substituting Eq. (5) into Eq. (4)

$H=\hbar\Omega^{\prime}_{1}(t)\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}+\hbar\Omega^{\prime}_{2}(t)\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2},$
(6)

where $\Omega_{j}^{\prime}(t)=\Omega_{j}+g_{j}s(t)$ is the modified resonant
frequency of the $j-th$ quantum harmonic oscillator $\hat{b}_{j}$ for $j=1,2$.
With the Hamiltonian given by Eq. (6), we now have the system master equation,

$\begin{split}\dot{\rho}=&i\left[\rho,\hbar\Omega_{1}^{\prime}(t)\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}+\hbar\Omega_{2}^{\prime}(t)\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2}\right]\\\
&+\sum_{j=1,2}\left[\Gamma_{j}[n_{j,{\rm
th}}(t)+1]\left(2\hat{\tilde{b}}_{j}\rho\hat{\tilde{b}}_{j}^{{\dagger}}-\hat{\tilde{b}}_{j}^{{\dagger}}\hat{\tilde{b}}_{j}\rho-\rho\hat{\tilde{b}}_{j}^{{\dagger}}\hat{\tilde{b}}_{j}\right)\right]\\\
&+\sum_{j=1,2}\left[\Gamma_{j}n_{j,{\rm
th}}(t)\left(2\hat{\tilde{b}}_{j}^{{\dagger}}\rho\hat{\tilde{b}}_{j}-\hat{\tilde{b}}_{j}\hat{\tilde{b}}_{j}^{{\dagger}}\rho-\rho\hat{\tilde{b}}_{j}\hat{\tilde{b}}_{j}^{{\dagger}}\right)\right],\end{split}$
(7)

where $\Gamma_{j}$ is the damping rate of the $j-th$ quantum harmonic
oscillator $b_{j}$, and its mean thermal photon (phonon) excitation number
$n_{j,{\rm th}}(t)=\exp{[{\hbar\Omega_{j}^{\prime}}(t)/{\kappa_{\rm
B}T_{j}}-1]^{-1}}$ is determined by the temperature $T_{j}$ and the effective
resonant frequency $\Omega_{j}^{\prime}(t)$. Here, $\kappa_{\rm B}$ is the
Boltzmann constant. From the master equation, we can then obtain the equations
of motions for
$\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}\rangle$,
$\langle\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2}\rangle$,
$\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{2}\rangle$,
$\langle\hat{\tilde{b}}_{1}\hat{\tilde{b}}_{2}\rangle$,
$\langle\hat{\tilde{b}}_{1}^{2}\rangle$, and
$\langle\hat{\tilde{b}}_{2}^{2}\rangle$ by applying
${\langle\dot{\hat{o}}\rangle}={\rm Tr}(\rho\hat{o})$ for the operator
$\hat{o}$,

$\displaystyle\frac{d\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}\rangle}{dt}=$
$\displaystyle-\Gamma_{1}\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}\rangle+\Gamma_{1}n_{j,\rm
th}[\Omega_{1}^{\prime}(t)],$ (8a)
$\displaystyle\frac{d\langle\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2}\rangle}{dt}=$
$\displaystyle-\Gamma_{2}\langle\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2}\rangle+\Gamma_{2}n_{j,\rm
th}[\Omega_{2}^{\prime}(t)],$ (8b)
$\displaystyle\frac{d\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{2}\rangle}{dt}=$
$\displaystyle-i[-\Omega_{1}^{\prime}(t)+\Omega_{2}^{\prime}(t)]\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{2}\rangle-\frac{\Gamma_{1}+\Gamma_{2}}{2}\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{2}\rangle,$
(8c)
$\displaystyle\frac{d\langle\hat{\tilde{b}}_{1}\hat{\tilde{b}}_{2}\rangle}{dt}=$
$\displaystyle-i[\Omega_{1}^{\prime}(t)+\Omega_{2}^{\prime}(t)]\langle\hat{\tilde{b}}_{1}\hat{\tilde{b}}_{2}\rangle-\frac{\Gamma_{1}+\Gamma_{2}}{2}\langle\hat{\tilde{b}}_{1}\hat{\tilde{b}}_{2}\rangle,$
(8d) $\displaystyle\frac{d\langle\hat{\tilde{b}}_{1}^{2}\rangle}{dt}=$
$\displaystyle[-2i\Omega_{1}^{\prime}(t)-\Gamma_{1}]\langle\hat{\tilde{b}}_{1}^{2}\rangle,$
(8e) $\displaystyle\frac{d\langle\hat{\tilde{b}}_{2}^{2}\rangle}{dt}=$
$\displaystyle[-2i\Omega_{2}^{\prime}(t)-\Gamma_{2}]\langle\hat{\tilde{b}}_{2}^{2}\rangle.$
(8f)

One can easily find that the values of
$\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{2}\rangle$,
$\langle\hat{\tilde{b}}_{1}\hat{\tilde{b}}_{2}\rangle$,
$\langle\hat{\tilde{b}}_{1}^{2}\rangle$, and
$\langle\hat{\tilde{b}}_{2}^{2}\rangle$ decay to zero since they are coupling
to dissipation but not subjecting to any driving. Here, for the non-zero terms
$\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}\rangle$ and
$\langle\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2}\rangle$, their
dynamics are dominated by $n_{1,\rm th}(t)$ and $n_{2,\rm th}(t)$,
respectively, where $n_{1,\rm th}(t)$ $[n_{2,\rm th}(t)]$ is known as a
function of the modified mechanical frequency $\Omega_{1}^{\prime}(t)$
[$\Omega_{2}^{\prime}(t)$].

###  III.2 Quantum synchronization of dissipative harmonic oscillators

Recall the definition of quantum synchronization discussed in Sec. II, where
the complete synchronization of two quantum harmonic oscillators $\hat{b}_{1}$
and $\hat{b}_{2}$ is achieved if their standard deviations [$\sigma_{1,x}(t)$,
$\sigma_{1,p}(t)$] and [$\sigma_{2,x}(t)$, $\sigma_{2,p}(t)$] satisfy the
conditions: $\lim_{t\rightarrow\infty}[\sigma_{1,x}(t)-\sigma_{2,x}(t)]=0$ and
$\lim_{t\rightarrow\infty}[\sigma_{1,p}(t)-\sigma_{2,p}(t)]=0$. Here,
$\sigma_{j,x}$ and $\sigma_{j,p}$ take the forms

$\sigma_{j,x}=\sqrt{\langle\hat{x}_{j}^{2}\rangle-\langle\hat{x}_{j}\rangle^{2}},\,\,\,\,\,\sigma_{j,p}=\sqrt{\langle\hat{p}_{j}^{2}\rangle-\langle\hat{p}_{j}\rangle^{2}},j=1,2.$
(9)

By applying the relations
$\hat{x}_{j}=({\hat{b}_{j}+\hat{b}^{\dagger}_{j}})/{\sqrt{2}}$ and
$\hat{p}_{j}=-i({\hat{b}_{j}-\hat{b}^{\dagger}_{j}})/{\sqrt{2}}$, and
separating $\hat{b}_{j}$ into a classical mean value and a quantum part:
$\hat{b}_{j}=\langle\hat{b}_{j}\rangle+\hat{\tilde{b}}_{j}$, the standard
deviations $\sigma_{j,x}$ and $\sigma_{j,p}$ of the quantum harmonic
oscillator $\hat{b}_{j}$ can be rewritten as

$\displaystyle\sigma_{j,x}$
$\displaystyle=\sqrt{\frac{1}{2}+\langle\hat{\tilde{b}}_{j}^{\dagger}\hat{\tilde{b}}_{j}\rangle+{\rm
Re}[\langle\hat{\tilde{b}}_{j}^{2}\rangle]},$ (10a)
$\displaystyle\sigma_{j,p}$
$\displaystyle=\sqrt{\frac{1}{2}+\langle\hat{\tilde{b}}_{j}^{\dagger}\hat{\tilde{b}}_{j}\rangle-{\rm
Re}[\langle\hat{\tilde{b}}_{j}^{2}\rangle]},j=1,2.$ (10b)

It can be seen from Eq. (10) that both $\sigma_{j,x}$ and $\sigma_{j,p}$ are
functions of $\langle\hat{\tilde{b}}_{j}^{\dagger}\hat{\tilde{b}}_{j}\rangle$
and $\langle\hat{\tilde{b}}_{j}^{2}\rangle$, thus we have new conditions for
quantum complete synchronization

$\lim_{t\rightarrow\infty}{e}_{\rm n_{b}}(t)=0,\ \ \ \ \
\lim_{t\rightarrow\infty}{e}_{\rm b^{2}}(t)=0,$ (11)

where ${e}_{\rm
n_{b}}=\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}\rangle-\langle\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2}\rangle$
and ${e}_{\rm
b^{2}}=\langle\hat{\tilde{b}}_{1}^{2}\rangle-\langle\hat{\tilde{b}}_{2}^{2}\rangle$
stand for the synchronization differences. Also, the equations of motion for
$\langle\hat{\tilde{b}}_{j}^{\dagger}\hat{\tilde{b}}_{j}\rangle$ and
$\langle\hat{\tilde{b}}_{j}^{2}\rangle$ can be found in Eq. (8)

$\displaystyle{d\langle\hat{\tilde{b}}_{j}^{\dagger}\hat{\tilde{b}}_{j}\rangle}/dt$
$\displaystyle=-\Gamma_{j}\langle\hat{\tilde{b}}_{j}^{\dagger}\hat{\tilde{b}}_{j}\rangle+\Gamma_{j}n_{j,\rm
th}[\Omega^{\prime}_{j}(t)],$ (12a) $\displaystyle
d{\langle\hat{\tilde{b}}_{j}^{2}\rangle}/dt$
$\displaystyle=-[2i\Omega_{j}^{\prime}(t)+\Gamma_{j}]\langle\hat{\tilde{b}}_{j}^{2}\rangle,j=1,2.$
(12b)

For complete synchronization, note that all the parameters of two quantum
harmonic oscillators are required to be identical:
$\Gamma_{1}=\Gamma_{2}=\Gamma$, $g_{1}=g_{2}=g$, and
$\Omega_{1}=\Omega_{2}=\Omega$. Then, one can easily obtain the equations of
motion for the synchronization differences from Eq. (12)

$\displaystyle\dot{e}_{\rm n_{b}}$ $\displaystyle=-\Gamma{e}_{\rm n_{b}},$
(13a) $\displaystyle\dot{e}_{\rm b^{2}}$
$\displaystyle=-[2i\Omega^{\prime}(t)+\Gamma]{e}_{\rm b^{2}}.$ (13b)

To check if the quantum synchronization conditions provided by Eq. (11) can be
satisfied, we analytically solve Eq. (13) and obtain the solutions: ${e}_{\rm
n_{b}}(t)=\exp(-\Gamma t){e}_{\rm n_{b}}(0)$ and ${e}_{\rm
b^{2}}(t)=\exp{(-2\Gamma
t)}\exp{\int^{t}_{0}{[-2i\Omega^{\prime}(t^{\prime})]}dt^{\prime}}e_{\rm
b^{2}}(0)$. We find that both of the synchronization differences ${e}_{\rm
n_{b}}$ and ${e}_{\rm b^{2}}$ converge to zero as $t\to\infty$ when
$\Gamma>0$. It thus can be concluded that the quantum synchronization led by
this approach is asymptotically stable when the quantum harmonic oscillators
are subject to dissipation.

The above discussion has demonstrated that the complete synchronization of two
quantum harmonic oscillators can be achieved via dissipative mechanisms
induced by the environment. The stability of the complete quantum
synchronization induced by the environment has implied that the quantum
synchronization can be also realized in chaotic regimes by the same mechanism
to be discussed in the next section.

Figure 3: (color online) An optomechanical setup for quantum synchronization
of two mechanical resonators $\hat{b}_{1}$ and $\hat{b}_{2}$ in the framework
of APD configuration.

##  IV A quantum synchronization model: periodic and chaotic motions

In this section, we study the implementation of the above quantum
synchronization model with an experimentally accessible optomechanical setup.
We numerically show that complete synchronization can be achieved in this mode
consisting of two quantum mechanical resonators. Interestingly, this simple
model actually allows us to show that the quantum synchronization is robust in
either periodic or chaotic regimes Strunz et al. (1999); Naghiloo et al.
(2017); Mourik et al. (2018); de Almeida (1988); Ullmo (2008); Wright and
Weaver (2010); Beenakker (1997); Stockmann (1999); Haake (1991); Riser et al.
(2017); Neill et al. (2016); Słomczyński and Życzkowski (1994); Zurek and Paz
(1995); Chirikov (1995); Kowalewska-Kudłaszyk et al. (2008); Geszti (2019,
2018); Cornelius et al. (2022); Nakamura (1993); Heller (2018); Gutzwiller
(1990); Heller and Tomsovic (1993); Zurek (1998); Habib et al. (1998);
Karkuszewski et al. (2002); Zurek (2003); Bhattacharya et al. (2000); Xu et
al. (2019).

As shown in Fig. 3, The setup consists of three components: an optomechanical
system (left hand) and two quadratic-coupling optomechanical systems (right
hand). The two quantum mechanical resonators ($\hat{b}_{1}$ and $\hat{b}_{2}$)
to be synchronized are distributed in the two separated quadratic-coupling
optomechanical systems (right hand), respectively. The left-hand
optomechanical system ($\alpha_{c}$, $\beta_{c}$) is strongly driven and thus
can be treated classically. It outputs a classical field that acts as the
inputs of the cavity fields ($\alpha_{1}$ and $\alpha_{2}$). The quantum
mechanical modes ($\hat{b}_{1}$ and $\hat{b}_{2}$) are modulated by the left-
hand optomechanical system ($\alpha_{c}$, $\beta_{c}$) via the quadratic
coupling with the cavity fields ($\alpha_{1}$ and $\alpha_{2}$). These form an
APD configuration that will lead to complete synchronization of the two
quantum mechanical resonators ($\hat{b}_{1}$ and $\hat{b}_{2}$). Here, the
Hamiltonian of the total system reads

$\begin{split}H_{\rm
total}=&\Delta_{c}\hat{a}_{c}^{\dagger}\hat{a}_{c}+\Omega_{c}{\hat{b}_{c}}^{\dagger}\hat{b}_{c}+g_{c}\hat{a}_{c}^{\dagger}\hat{a}_{c}(\hat{b}_{c}^{\dagger}+\hat{b}_{c})+\varepsilon_{c}(\hat{a}_{c}^{\dagger}+\hat{a}_{c})\\\
&+\Delta_{1}\hat{a}_{1}^{\dagger}\hat{a}_{1}+\Delta_{2}\hat{a}_{2}^{\dagger}\hat{a}_{2}+\Omega_{2}{\hat{b}_{1}}^{\dagger}\hat{b}_{1}+\Omega_{2}{\hat{b}_{2}}^{\dagger}\hat{b}_{2}\\\
&+g_{1}\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{b}_{1}^{\dagger}\hat{b}_{1}+g_{2}\hat{a}_{2}^{\dagger}\hat{a}_{2}\hat{b}_{2}^{\dagger}\hat{b}_{2},\end{split}$
(14)

where $\Delta_{k}=\omega_{k}-\omega_{k,d}$ is the detuning between the
resonant frequency $\omega_{k}$ and the external driving $\omega_{k,d}$ of the
cavity mode $\hat{a}_{k}$ for $k=1,2,c$; and its driving strength is denoted
by $\varepsilon_{k}$. Here, the resonant frequency and the damping rate of the
mechanical mode $\hat{b}_{k}$ are $\Omega_{k}$ and $\Gamma_{k}$, respectively;
while $g_{k}$ is the optomechanical coupling strength between the cavity mode
$\hat{a}_{k}$ and the mechanical resonator $\hat{b}_{k}$.

Here, the equations of motion of each cavity (mechanical) mode are described
by the Langevin equations

$\displaystyle\dot{\hat{a}}_{c}$
$\displaystyle=-i\Delta_{c}\hat{a}_{c}-ig_{c}\hat{a}_{c}(\hat{b}_{c}^{\dagger}+\hat{b}_{c})-\frac{\gamma_{c}}{2}\hat{a}_{c}+\varepsilon_{c}-\sqrt{\gamma_{c}}\,\hat{a}_{c,\rm
in},\;$ (15a) $\displaystyle\dot{\hat{b}}_{c}$
$\displaystyle=-i\Omega_{c}\hat{b}_{c}-ig_{c}\hat{a}_{c}^{\dagger}\hat{a}_{c}-\frac{\Gamma_{\\!c}}{2}\hat{b}_{c}-\sqrt{\Gamma_{\\!c}}\,\hat{b}_{c,\rm
in},\;$ (15b) $\displaystyle\dot{\hat{a}}_{1}$
$\displaystyle=-i\Delta_{1}\hat{a}_{1}-\frac{\gamma_{1}}{2}\hat{a}_{1}-ig_{1}\hat{a}_{1}\hat{b}_{1}^{\dagger}\hat{b}_{1}-\sqrt{\gamma_{1}\gamma_{c}}\,\hat{a}_{c}-\sqrt{\gamma_{1}}\hat{a}_{1,\rm
in},\;$ (15c) $\displaystyle\dot{\hat{a}}_{2}$
$\displaystyle=-i\Delta_{2}\hat{a}_{2}-\frac{\gamma_{2}}{2}\hat{a}_{2}-ig_{2}\hat{a}_{2}\hat{b}_{2}^{\dagger}\hat{b}_{2}-\sqrt{\gamma_{2}\gamma_{c}}\,\hat{a}_{c}-\sqrt{\gamma_{2}}\hat{a}_{2,\rm
in},\;$ (15d) $\displaystyle\dot{\hat{b}}_{1}$
$\displaystyle=-i\Omega_{1}\hat{b}_{1}-\frac{\Gamma_{1}}{2}\hat{b}_{1}-ig_{1}\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{b}_{1}^{\dagger}-\sqrt{\Gamma_{1}}\hat{b}_{1,\rm
in},\;$ (15e) $\displaystyle\dot{\hat{b}}_{2}$
$\displaystyle=-i\Omega_{2}\hat{b}_{2}-\frac{\Gamma_{2}}{2}\hat{b}_{2}-ig_{2}\hat{a}_{2}^{\dagger}\hat{a}_{2}\hat{b}_{2}^{\dagger}-\sqrt{\Gamma_{2}}\hat{b}_{2,\rm
in},\;$ (15f)

where $\gamma_{k}$ and $\hat{a}_{k,\rm in}$ are the damping rate and the input
of the optical cavity $\hat{a}_{k}$, and $\hat{b}_{k,\rm in}$ and
$\Gamma_{\\!k}$ are the input and the damping rate of the mechanical mode
$\hat{b}_{k}$ for $k=1,2,c$.

In this setup, the optomechanical resonator ($\hat{a}_{c}$, $\hat{b}_{c}$) and
the cavity modes ($\hat{a}_{1}$ and $\hat{a}_{2}$) can be considered
classically. By replacing the quantum operators with their classical averages
in Eq. (15): $\alpha_{1}=\langle\hat{a}_{1}\rangle$,
$\alpha_{2}=\langle\hat{a}_{2}\rangle$,
$\alpha_{c}=\langle\hat{a}_{c}\rangle$, and
$\beta_{c}=\langle\hat{b}_{c}\rangle$, we can then obtain the equations of
motions for the classical parts

$\displaystyle\dot{\alpha}_{c}$
$\displaystyle=-i\Delta_{c}\alpha_{c}-\frac{\gamma_{c}}{2}\alpha_{c}-ig_{c}\alpha_{c}(\beta_{c}+\beta_{c}^{*})+\varepsilon_{c},$
(16a) $\displaystyle\dot{\beta}_{c}$
$\displaystyle=\left(-i\Omega_{c}-\frac{\Gamma_{c}}{2}\right)\beta_{c}-ig_{c}|\alpha_{c}|^{2},$
(16b) $\displaystyle\dot{\alpha}_{1}$
$\displaystyle=-i\Delta_{1}\alpha_{1}-\frac{\gamma_{1}}{2}\alpha_{1}-\sqrt{\gamma_{1}\gamma_{c}}{\alpha}_{c}+\varepsilon_{1},$
(16c) $\displaystyle\dot{\alpha}_{2}$
$\displaystyle=-i\Delta_{2}\alpha_{2}-\frac{\gamma_{2}}{2}\alpha_{2}-\sqrt{\gamma_{2}\gamma_{c}}{\alpha}_{c}+\varepsilon_{2}.$
(16d)

Note that the cavity modes $\alpha_{1}$ and $\alpha_{2}$ are not affected by
the mechanical resonators $\hat{b}_{1}$ and $\hat{b}_{2}$. This is because the
latter does not contain driving terms, as such the classical averages
$\beta_{1}$ and $\beta_{2}$ converge to zero as the time $t$ increases.

Now, we focus on the dynamical evolution of the quantum mechanical resonator
$\hat{b}_{1}$ and $\hat{b}_{2}$. Since the optomechanical resonator
($\hat{a}_{c}$, $\hat{b}_{c}$) and the cavity modes ($\hat{a}_{1}$ and
$\hat{a}_{2}$) treated classically are omitted in the total Hamiltonian [Eq.
(14)], we have the effective system Hamiltonian,

$\tilde{H}_{\rm
eff}=\hbar\Omega^{\prime}_{1}(t)\hat{b}_{1}^{\dagger}\hat{b}_{1}+\hbar\Omega^{\prime}_{2}(t)\hat{b}_{2}^{\dagger}\hat{b}_{2},$
(17)

where $\Omega^{\prime}_{j}=\Omega_{j}+g_{j}|\alpha_{j}|^{2}$ is the modified
mechanical frequency due to the optomechanical coupling with the classical
optical field $\alpha_{j}$ for $j=1,2$. Here, the cavity mode $\alpha_{1}$
($\alpha_{2}$) links both the classical optomechanical system ($\alpha_{c}$,
$\beta_{c}$) and the quantum mechanical mode $\hat{b}_{1}$ ($\hat{b}_{2}$)
together. The chaos generated by the optomechanical system ($\alpha_{c}$,
$\beta_{c}$), is thus transferred into the quantum mechanical resonators
$\hat{b}_{1}$ and $\hat{b}_{2}$.

Remind that the dynamics of the quantum mechanical mode $\hat{b}_{1}$
($\hat{b}_{2}$), is described by the temporal evolution of its standard
deviations ($\sigma_{1,x}$, $\sigma_{1,p}$) [($\sigma_{2,x}$,
$\sigma_{2,p}$)], which are given in Eq. (10). Specifically, in this
quadratic-coupling optomechanical setting, the mean value of the mechanical
resonator $\hat{b}_{1}$ ($\hat{b}_{2}$) is always zero:
$\langle\hat{b}_{1}\rangle=0$ ($\langle\hat{b}_{2}\rangle=0$). Thus, the
standard deviations ($\sigma_{j,x}$, $\sigma_{j,p}$) for the quantum
mechanical mode $\hat{b}_{1}$ ($\hat{b}_{2}$) can be rewritten as

$\displaystyle\sigma_{j,x}$
$\displaystyle=\sqrt{\frac{1}{2}+\langle\hat{b}_{j}^{\dagger}\hat{b}_{j}\rangle+{\rm
Re}[\langle\hat{b}_{j}^{2}\rangle]},$ (18a) $\displaystyle\sigma_{j,p}$
$\displaystyle=\sqrt{\frac{1}{2}+\langle\hat{b}_{j}^{\dagger}\hat{b}_{j}\rangle-{\rm
Re}[\langle\hat{b}_{j}^{2}\rangle]},j=1,2,$ (18b)

where the equations of motions for
$\langle\hat{b}_{1}^{\dagger}\hat{b}_{1}\rangle$,
$\langle\hat{b}_{2}^{\dagger}\hat{b}_{2}\rangle$,
$\langle\hat{b}_{1}^{\dagger}\hat{b}_{2}\rangle$,
$\langle\hat{b}_{1}\hat{b}_{2}\rangle$, $\langle\hat{b}_{1}^{2}\rangle$, and
$\langle\hat{b}_{2}^{2}\rangle$ are given by

$\displaystyle\frac{d\langle\hat{b}_{1}^{\dagger}\hat{b}_{1}\rangle}{dt}=$
$\displaystyle-\Gamma_{1}\langle\hat{b}_{1}^{\dagger}\hat{b}_{1}\rangle+\Gamma_{1}n_{1,\rm
th}[\Omega_{1}^{\prime}(t)],$ (19a)
$\displaystyle\frac{d\langle\hat{b}_{2}^{\dagger}\hat{b}_{2}\rangle}{dt}=$
$\displaystyle-\Gamma_{2}\langle\hat{b}_{2}^{\dagger}\hat{b}_{2}\rangle+\Gamma_{2}n_{2,\rm
th}[\Omega_{2}^{\prime}(t)],$ (19b)
$\displaystyle\frac{d\langle\hat{b}_{1}^{\dagger}\hat{b}_{2}\rangle}{dt}=$
$\displaystyle-i[-\Omega_{1}^{\prime}(t)+\Omega_{2}^{\prime}(t)]\langle\hat{b}_{1}^{\dagger}\hat{b}_{2}\rangle-\frac{\Gamma_{1}+\Gamma_{2}}{2}\langle\hat{b}_{1}^{\dagger}\hat{b}_{2}\rangle,$
(19c) $\displaystyle\frac{d\langle\hat{b}_{1}\hat{b}_{2}\rangle}{dt}=$
$\displaystyle-i[\Omega_{1}^{\prime}(t)+\Omega_{2}^{\prime}(t)]\langle\hat{b}_{1}\hat{b}_{2}\rangle-\frac{\Gamma_{1}+\Gamma_{2}}{2}\langle\hat{b}_{1}\hat{b}_{2}\rangle,$
(19d) $\displaystyle\frac{d\langle\hat{b}_{1}^{2}\rangle}{dt}=$
$\displaystyle[-2i\Omega_{1}^{\prime}(t)-\Gamma_{1}]\langle\hat{b}_{1}^{2}\rangle,$
(19e) $\displaystyle\frac{d\langle\hat{b}_{2}^{2}\rangle}{dt}=$
$\displaystyle[-2i\Omega_{2}^{\prime}(t)-\Gamma_{2}]\langle\hat{b}_{2}^{2}\rangle.$
(19f)

One can easily find that the values of
$\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{2}\rangle$,
$\langle\hat{\tilde{b}}_{1}\hat{\tilde{b}}_{2}\rangle$,
$\langle\hat{\tilde{b}}_{1}^{2}\rangle$, and
$\langle\hat{\tilde{b}}_{2}^{2}\rangle$ decay to zero as the time $t$
increases. For the inhomogeneous equations
$\langle\hat{\tilde{b}}_{1}^{\dagger}\hat{\tilde{b}}_{1}\rangle$
($\langle\hat{\tilde{b}}_{2}^{\dagger}\hat{\tilde{b}}_{2}\rangle$), the time
evolution of these terms will be sensible to $n_{1,\rm th}$ ($n_{2,\rm th}$),
where $n_{j,\rm th}(t)=\exp{[{\hbar\Omega_{j}^{\prime}}(t)/{\kappa_{\rm
B}T}-1]^{-1}}$ is the mean thermal phonon number at the temperature $T$ and
the modified resonant frequency $\Omega_{j}^{\prime}(t)$ in this setting.

Figure 4: (color online) The phase space orbits of the quantum mechanical
resonator $\hat{b}_{1}$ for different classical optical detunings
$\Delta_{c}/\Omega_{c}$: (a) $-0.4$, (b) $-0.6$, (c) $-0.85$, and (d) $-0.95$.
Other parameters are set as: $\gamma_{c}/\Omega_{c}=1$,
$g_{c}/\Omega_{c}=0.001$, $\Gamma_{c}/\Omega_{c}=0.001$,
$\Omega_{c}/2\pi=1~{}\rm GHz$, $\varepsilon_{c}/\Omega_{c}=418$,
$\Delta_{1}/\Omega_{1}=-2$, $\gamma_{1}/\Omega_{1}=1$,
$\varepsilon_{1}/\Omega_{1}=0$, $g_{1}/\Omega_{1}=0.001$,
$\Gamma_{1}/\Omega_{1}=10$, $\Omega_{1}/2\pi=0.01~{}\rm GHz$, and
$T=0.002~{}\rm k$.

Also, since $\lim_{t\rightarrow\infty}{\rm
Re}[\langle\hat{b}_{j}^{2}\rangle]=0$ ($j=1,2$), we have the relation
$\sigma_{j,x}(t)=\sigma_{j,p}(t)$ from Eq. (18), the condition for quantum
synchronization in this setting is simplified as

$\lim_{t\rightarrow\infty}[\sigma_{1,x}(t)-\sigma_{2,x}(t)]=0.$ (20)

Here, the temporal evolutions of the standard deviations $\sigma_{j,x}$ and
$\sigma_{j,p}$ are determined by
$\langle\hat{b}_{j}^{\dagger}\hat{b}_{j}\rangle$, whose equations of motion
are governed by

$\frac{d\langle\hat{b}_{j}^{\dagger}\hat{b}_{j}\rangle}{dt}=-\Gamma_{j}\langle\hat{b}_{j}^{\dagger}\hat{b}_{j}\rangle+\Gamma_{j}n_{j,\rm
th}[\Omega_{j}^{\prime}(t)],j=1,2.$ (21)

To achieve the complete synchronization, the parameters and the classical
inputs of the two quantum mechanical modes $\hat{b}_{1}$ and $\hat{b}_{2}$ are
assumed to be identical: $\Omega_{1}=\Omega_{2}$, $\Gamma_{1}=\Gamma_{2}$, and
$\alpha_{1}(t)\equiv\alpha_{2}(t)$. Below we will show numerically that
quantum synchronization can be reached in $\hat{b}_{1}$ and $\hat{b}_{2}$ for
various settings.

In our simulations, we first prepare the classical controller ($\alpha_{c}$,
$\beta_{c}$) to four different regimes by adjusting its detuning
$\Delta_{c}/\Omega_{c}$: 1-period ($-0.4$), 2-period ($-0.6$), 4-period
($-0.85$), and chaos ($-0.95$). For each case, the initial conditions are set
to be different: $\sigma_{1,x}=\sqrt{1.5}$ and $\sigma_{2,x}=\sqrt{10.5}$. The
corresponding phase space orbits of the quantum mechanical resonator
$\hat{b}_{1}$ are shown in Fig. 4, we find that the quantum dynamics is
dominated by the classical controller ($\alpha_{c}$, $\beta_{c}$), which was
yet presented in Ref. Yang et al. . Here, the phase portraits of the quantum
system are reconstructed from the time-delayed coordinates
$\sigma_{1,x}(\tau),\sigma_{1,x}(2\tau),...,\sigma_{1,x}(N\tau)$, where
$\tau=0.3~{}{\rm ns}$.

Figure 5: (color online) The standard deviations $\sigma_{1,x}$ and
$\sigma_{2,x}$ of the quantum mechanical resonators $\hat{b}_{1}$ and
$\hat{b}_{1}$ for different quantum motions given in Fig. 4. Here, the
parameters are set as: $\Delta_{1}/\Omega_{1}=\Delta_{2}/\Omega_{2}=-2$,
$\gamma_{1}/\Omega_{1}=\gamma_{2}/\Omega_{2}=1$,
$\varepsilon_{1}/\Omega_{1}=\varepsilon_{2}/\Omega_{2}=0$,
$g_{1}/\Omega_{1}=g_{2}/\Omega_{2}=0.001$,
$\Gamma_{1}/\Omega_{1}=\Gamma_{2}/\Omega_{2}=10$,
$\Omega_{1}/2\pi=\Omega_{2}/2\pi=0.01~{}\rm GHz$. Other parameters are the
same as in Fig. 4.

The complete synchronization of two quantum mechanical resonators
$\hat{b}_{1}$ and $\hat{b}_{2}$ are presented in Fig. 5. For each quantum
regime given in Fig. 4, the values of $\sigma_{1,x}(t)$ and $\sigma_{2,x}(t)$
merge together as the time increases whereas they start from different initial
conditions. The complete synchronization is shown to be robust for different
quantum motions. Above numerical results are consistent with the analytic
proof given in Sec. III. The quantum synchronization is shown to be realizable
in both stable and unstable regimes, e.g., quantum chaos in Fig. 5(d).

Figure 6: (color online) Average synchronization error $E_{\rm avg}$ for
different mismatched parameters $\Delta_{\Gamma}$ and $\Delta_{G}$. Here,
$\Delta_{c}/\Omega_{c}=-1$, $g_{1}=10$ MHz, $\Gamma_{1}=0.15$ GHz, and
$\Omega_{1}=\Omega_{2}=10$ MHz. Other parameters are the same as in Fig. 4.

Moreover, we consider the cases that two quantum mechanical modes
$\hat{b}_{1}$ and $\hat{b}_{2}$ have different parameters. To be more
specific, here, we define the mismatched damping rate and mismatched resonant
frequency as $\Delta_{\Gamma}=(\Gamma_{1}-\Gamma_{2})/\Gamma_{1}$ and
$\Delta_{G}=(G_{1}-G_{2})/G_{1}$. Then, we introduce the average
synchronization error $E_{\rm avg}$ to measure the effect brought by these
mismatched parameters: $E_{\rm avg}=\|\int^{\infty}_{t_{0}}{\bf
e}(t)dt/\int^{\infty}_{t_{0}}\sigma_{1,x}(t)dt\|$, where $t_{0}$ is the
initial time. As shown in Fig. 6, the values of $E_{\rm avg}$ are plotted in
the $\Delta_{\Gamma}\mbox{-}\Delta_{G}$ plane and characterized by different
colors. It can be seen that $E_{\rm avg}$ is less than $0.01$ even when the
mismatched damping rate $\Delta_{\Gamma}$ is as high as $0.4$; and $E_{\rm
avg}$ is $0.03$ when the mismatched coupling strength $\Delta_{G}$ reaches the
value $0.1$. Here, it is shown that the complete synchronization of two
quantum mechanical modes is still robust for the mismatched parameters
$\Delta_{\Gamma}$ and $\Delta_{g}$.

##  V Conclusions and discussions

In this paper, we study the synchronization of a continuous-variable system
consisting of two quantum harmonic oscillators coupled to dissipative
environments. We show that the Active-Passive Decomposition configuration
defined in the classical dynamical systems plays a very important role in the
quantum regimes where the quantum synchronization can be realized. For the
physical models under consideration, it is proved that the quantum
synchronization is asymptotically stable if the quantum systems are subject to
dissipation. Moreover, as an example, an experimentally accessible model based
on an optomechanical setup is used to illustrate our approach on the quantum
synchronization process defined in this paper. The numerical simulations have
clearly indicated that complete synchronization can be achieved and is robust
to small parameter mismatches. It is shown that this quantum synchronization
approach is not only robust to limit circles, but also chaotic motions. It is
desirable to consider the quantum synchronization in different quantum open
systems such as dephasing noise, classical noises and colored noises, which
will be left for future publications.

###### Acknowledgements.

This project is partly supported by ART020-Quantum Technologies Project.

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