1. Introduction

The optomechanical system is believed as an exceedingly excellent platform in quantum information and quantum optics [1], in which a variety of physical phenomena are observed such as phonon laser [2,3], quantum transducers [4,5], blockade effect [6], or topological energy transfer [7]. Most recently, mechanical oscillators have come to the forefront due to their large coherence time and interaction with photons in cavity optomechanical systems [8,9]. Additionally, ground-state cooling of macroscopic mechanical oscillators is a crucial task. In order to perform quantum manipulation on mechanical oscillators such as mechanical squeezing [10,11], entanglement [12], and back-action evading measurements below the standard quantum limit [13–15], the oscillators need to be precooled to their ground states. Until now, tremendous theoretical research has been proposed to cool the mechanical oscillator to the ground state such as side-band cooling [16–19], feedback cooling scheme [20], cavity-assisted cooling [21], and transient cooling [22]. Moreover, there has been significant progress in achieving ground-state cooling experimentally in the optomechanical system [23].

In recent theoretical work, abundant proposals have been presented to achieve multiple mechanical oscillators cooling [24–26]. Sommer et al. [24] have provided partial optomechanical refrigeration of multimode resonators by cold-damping technique. Lai et al. [25] have proposed a domino-cooling method to realize simultaneous ground-state cooling of a coupled mechanical-resonator chain. Bai et al. [26] have proposed a scheme to achieve double-mechanical-oscillator cooling by introducing the frequency modulation and the bias gate voltage modulation. However, if multiple mechanical oscillators simultaneously couple to the same optical field and are identical or in the near-degenerate state, the simultaneous ground-state cooling can not be achieved due to the existence of the dark-mode effect [27–29]. Because the dark mode is decoupled from the optical field, meanwhile, there is no interaction between the dark mode and the bright mode due to the same frequencies of all oscillators, the simultaneous ground-state cooling can not be achieved. Nowadays a few kinds of research concentrate on overcoming this difficulty such as quantum reservoir engineering [30] , introducing a phase-dependent phonon-exchange interaction [31], cold-damping feedback [24,32], where the dark-mode effect is broken. Furthermore, if the difference between the two mechanical frequencies is not too small [27], they can be simultaneously cooled by the cavity field. Our group has proposed a scheme for the simultaneous ground-state cooling and synchronization of two mechanical oscillators by driving a nonlinear medium, where a frequency difference is modulated between two oscillators so that there is no dark-mode effect [33].

It is worth mentioning that quantum control has attracted tremendous attention in quantum physics [34–38]. The quantum Lyapunov control is particularly significant, where a specific Lyapunov function is selected to design the time-varying control field. According to the designed control field, the system will evolve to an expected target state. Quantum Lyapunov control has widespread applications in quantum physics, for example, generating the nonclassical states [39], studying synchronization [40], achieving strong mechanical squeezing [41], and applying to machine learning [42].

From the previous research, we can know that for the mechanical oscillators with different frequencies, simultaneous ground-state cooling can be achieved, in which the cooling progress is not affected by the dark-mode effect. However, for identical mechanical oscillators with the same frequency, this technique is not applicable. In [31], the dark-mode effect is broken through a loop-coupled optomechanical system, where the coupling between the two mechanical modes depends on the modulation of phase. Different from [31], we propose a scheme to achieve the simultaneous ground-state cooling of identical mechanical oscillators by utilizing the Lyapunov control. The time-variant voltage is applied to the identical oscillators, which has been already achieved experimentally. Due to the application of voltage, the frequencies are tunable so that the coupling between the bright mode and the dark mode can be reconstructed. After the cooling process, the modulated oscillators will return to the eigenfrequencies in the steady state. In order to illuminate the mechanism of the simultaneous ground-state cooling of the multiple oscillators, we deduce the evolution between bright and dark modes from decoupling to rebuilding the effective coupling in detail in the Appendix. We also study the steady-state average phonon number varying with the cavity field dissipation and the effective optomechanical coupling ratio of two mechanical oscillators. Except for achieving the simultaneous ground-state cooling of two mechanical oscillators, this proposal is generalized to multiple mechanical oscillators.

2. Method of quantum Lyapunov control

We introduce the general Lyapunov control method to find the control field [43]. The Hamiltonian of the quantum control system is $H=H_{0}+\sum _{n}f_{n}(t)H_{n}$, ($n=1, \ldots k$), where $H_{0}$ is the free Hamiltonian and $H_{n}$ is the control Hamiltonian. $f_{n}(t)$ is a time-dependent control field. The Lyapunov control function is defined as $V= \mathrm {Tr}$ $(\rho P)$, where $P$ is a positive and semidefinite Hermitian operator which can guarantee the mean value $V(t)\geq 0$. Applying the Liouville equation, the quantum system can be described as

Therefore, the derivative of control function $V$ is written as $\dot {V}=\sum _{n}f_{n}(t)T_{n}$, where we assume $[P,H_{0}]=0$, and $T_{n}=\mathrm {Tr}(-i\rho [P,H_{n}])$. If we would like to control the mechanical quantity $\langle P\rangle$ monotonously decreasing, that is, $\dot {V}<0$, we assume the control field $f_{n}(t)=-cT_{n}$. $c$ is a real positive constant so the derivative of the mean value can be written as $\dot {V}(t)=-\sum _{n}cT_{n}^{2}\leq 0$, which means that the derivative $\dot {V}\leq 0$ is always satisfied and the mean value $\langle P\rangle$ will decrease monotonically varying with the time evolution. Actually, the constant c plays a role in modulating the convergence of the control function so that the actual state will converge to the target state under the modulation of the control field $f_{n}(t)$.

3. Model and equation of motion

We consider an optomechanical system, where two mechanical oscillators simultaneously couple to a driven optical field with coupling constants $g_{1}$ and $g_{2}$ as shown in Fig. 1, where a time-dependent frequency modulation is applied to oscillator 2. Experimentally, the tunable frequency can be achieved by applying a gate voltage $V_{g}$ between the mechanical oscillator and the underlying electrode [44,45]. For instance, Barton et al. [44] have proposed that a graphene mechanical resonator comprises one end of a Fabry-Perot cavity via photothermal back-action. Owing to graphene’s low stiffness in combination with strong electrostatic coupling, the tunable frequency can be achieved by applying a modulated voltage $V_{g}$ between the graphene and the gate. Weber et al. [45] have proposed that a high-Q graphene mechanical resonator capacitively couples to a superconducting microwave cavity, where by applying a constant voltage to the graphene, the frequency of the resonator can be dramatically tuned.

 

Fig. 1. The schematic diagram of system. A time-dependent frequency modulation is applied to the oscillator 2, which can be modulated by a gate voltage $V_{g}$.

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The total Hamiltonian of the whole hybrid system is $H=H_{0}+H_{I}+H_{d}$ with

For generality, we introduce a fixed frequency $\omega _{2}$. But in our numerical calculation, we consider two mechanical oscillators are identical, $\omega _{2}=\omega _{1}$, and the initial value of controllable frequency $\omega _{r}(0)=\omega _{2}=\omega _{1}$. With the definition of position and momentum in Eq. (3), the Hamiltonian can be written as

The evolution of the system can be described by the Langevin equations as

Since the cavity field is driven by a strong classical field, we can write the operator $a=\alpha +\delta a$, $o_{j}={o_{j}}_{ss}+\delta o_{j}$ and $o\in (q,p)$, where $\alpha ~({o_{j}}_{ss})$ is a classical mean value of the cavity (mechanical) mode, and $\delta a~(\delta o_{j})$ is the fluctuation operator. For simplicity, hereafter we will omit $\delta$ and write the operator $\delta a$ ($\delta o_{j}$) as $a$ ($o_{j}$). The classical motion equations are

By introducing the quadrature operators $X_{a}=(a+a^{\dagger})/\sqrt {2}$, $Y_{a}=(a-a^{\dagger})/\sqrt {2}i$ and the corresponding noise terms $X_{a_{in}}=(a_{in}+a^{\dagger}_{in})/\sqrt {2}$, $Y_{a_{in}}=(a_{in}-a^{ \dagger}_{in})/\sqrt {2}i$, the linearized Langevin equations for the fluctuation operators can be compactly expressed as

Substituting the formal integral $u(t)$ into Eq. (9), we can obtain the derivative of the covariance matrix as

The derivative of $\bar {N}_{b_{j}}(t)$ can be obtained by solving Eq. (10). In the following calculations, the initial states of mechanical and optical modes considered are thermal equilibrium state and vacuum state so that $M(0)=\mathrm {Diag}[1/2,1/2,\bar {n}_{1}+1/2,\bar {n}_{1}+1/2, \bar {n} _{2}+1/2,\bar {n}_{2}+1/2]$.

4. Simultaneous cooling of two mechanical oscillators

Applying the procedure proposed in Section. 2, $\bar {N}_{b_{1}}(t)$ and $\bar {N}_{b_{2}}(t)$ are selected as the Lyapunov functions which satisfy $\bar {N}_{b_{j}}(t)>0$. According to Eqs. (9)–(11), the time derivatives of $\bar {N}_{b_{j}}(t)$ can be written as

We now analyze the correlations between the optical field and the mechanical oscillator, where $L_{1}=\sqrt {2}G_{1}(M_{14}+M_{41})$ and $L_{2}=\sqrt {2} G_{2}(M_{16}+M_{61})$ correspond to the first term in Eq. (12) and Eq. (13). In Fig. 2(a) and 2(b), we plot $L_{1}$ and $L_{2}$ as functions of $\kappa /\omega _{1}$ and $\omega _{1}t$, while as functions of $G/\omega _{1}$ and $\omega _{1}t$ for Fig. 2(c) and 2(d), where the parts surrounded by black line are greater than 0. We can see that in most areas, $L_{1}$ and $L_{2}$ are negative values within the proper parameters. Although in some regions the values are more than 0, these values are so small that they are negligible. That is because the correlations $L_{1}$ and $L_{2}$ are relatively small, and the selected effective optomechanical couplings are weak enough, the correlation terms can be omitted. Furthermore, the dissipation of mechanical oscillators is small enough that the thermal fluctuations of mechanical oscillators can be neglected, and the term $-\gamma _{1}M_{44}<0$ ($-\gamma _{2}M_{66}<0$) is always satisfied. In order to guarantee $\dot {\bar {N}}_{b_{2}}(t)\leq 0$, we choose

The control field in our work can be obtained by the simulation to achieve simultaneous ground-state cooling. $\dot {\bar {N}}_{b_{j}}(t)$ can be rewritten as

It is obvious that $\bar {N}_{b_{j}}(t)$ will decrease monotonically during evolution.

 

Fig. 2. (a) and (b) show $L_{1}$ and $L_{2}$ as functions of $ \omega _{1}t$ and $ \kappa / \omega _{1}$. (c) and (d) show $L_{1}$ and $L_{2}$ as functions of $ \omega _{1}t$ and $G/ \omega _{1}$, where $G_{1}=G_{2}=G$. The parts surrounded by the black line are greater than 0. For (a) and (b), $G_{1}=G_{2}=0.03 \omega _{1}$ while for (c) and (d), $ \kappa =0.1 \omega _{1}$. The other parameters are $\bar {n }_{1}=\bar {n}_{2}=100$, $ \gamma _{1}= \gamma _{2}=10^{-6} \omega _{1}$, $c=0.32$, $\Delta _{c}= \omega _{1}$ and $ \omega _{2}= \omega _{1}$.

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To understand the simultaneous ground-state cooling of two mechanical oscillators by Lyapunov control, we plot the time evolution of the average phonon numbers for oscillators 1 and 2 and the time-varying frequency of applying to oscillator 2, shown in Fig. 3. According to $\bar {n}_{j}=[{\rm {\ exp}}(\frac {\hbar \omega _{j}}{k_{B}T})-1]^{-1}$, although the heat phonon number varies with the frequency of the oscillator, the frequency-modulated time of the oscillator varies rapidly and the thermal equilibrium is assumed unchanged. For Fig. 3(a) and 3(b), $\omega _{r}(t)\equiv \omega _{2}=\omega _{1},$ two mechanical oscillators possess identical eigenfrequencies. The dark mode decouples with the bright mode so that the average phonon numbers of mechanical oscillators can not be simultaneously cooled. In Fig. 3(d), the frequency of oscillator 2 has been effectively modulated so that the average phonon numbers of two mechanical oscillators can be simultaneously cooled, which is shown in Fig. 3(c). In the subgraph of Fig. 3(c), we plot the average phonon number varying with the constant c in the steady state, where the oscillator 1 is selected as an example, and the other parameters are the same as Fig. 2. Different values of c affect the optimal cooling effect through the frequency $\omega _{r}(t)$. Within the range $(0.3,0.4)$, the best cooling effect is achieved at $c=0.32$. In our calculation, we will choose the specific value. It is worth noting that $\omega _{r}(0)=\omega _{r}(\infty )=\omega _{2}=\omega _{1}$, which means that the two oscillators are of the same eigenfrequency, and the Lyapunov control only takes action during the dynamical cooling process. In other words, by Lyapunov control, two identical oscillators can reach simultaneous ground-state cooling. Due to the bright mode coupling to the optical field as well as an effective coupling between the bright mode and the dark mode by Lyapunov control, a cascade cooling channel is formed. Therefore, the optical field can extract the thermal excitations from oscillator 1 to the vacuum optical field owing to the direct coupling, and the dark mode can be cooled through the effective coupling between the bright mode and dark mode. The detailed discussions can be found in Appendix A.

 

Fig. 3. (a) and (c) show the time evolution of the average phonon numbers for two oscillators while (b) and (d) show the tunable frequency of oscillator 2. Here, all the graphics are as the functions of $ \omega _{1}t$. But the subgraph of (c) shows the mean phonon number of oscillator 1 varying with the constant c in the steady state. For (a) and (b), $ \omega _{r}(t)\equiv 1$ without Lyapunov control while for (c) and (d), $ \omega _{r}(t)$ is tunable by Lyapunov control. Here, $G_{1}=G_{2}=0.03 \omega _{1}$, $ \kappa =0.1 \omega _{1}$, and the other parameters are the same as Fig. 2 .

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In Fig. 4, we plot the average phonon number in the steady state as the functions of $\kappa /\omega _{1}$ (Fig. 4(a)) and $G_{2}/G_{1}$ (Fig. 4(b)). From Fig. 4(a), we can see that the graphs of $\bar {N}_{b_{1}}$ and $\bar {N} _{b_{2}}$ coalesce well, and the mechanical oscillators can be simultaneously cooled at the proper parameters. The cooling effect has a range of tolerance to the cavity dissipation. When $\kappa /\omega _{1}\in (0.038,0.143)$, under the resolved-sideband regime, the mean phonon number can be cooled to the ground state. That is because there is competition between the cooling efficiency of the cavity field vacuum bath and the sideband condition [31]. In Ref. [33], a nonlinear medium is introduced to generate another heating path. The Stokes heating can be suppressed owing to the destructive interference produced by multiple heating paths so that the ground-state cooling can be achieved in the unresolved-sideband regime. If one would like to achieve simultaneous cooling in the current scheme, one can introduce a nonlinear medium. Fig. 4(b) shows that the optimal simultaneous cooling occurs when $G_{1}=G_{2}$, where all parameters are the same, and the system is symmetric-coupling. The efficiency of the cavity vacuum bath which extracts the thermal excitations from two mechanical oscillators is equivalent. It is obviously seen that the greater the strength between the oscillator and cavity field, the lower the average phonon numbers, which is shown in the locally magnified subgraphs of Fig. 4(b). In the deduction of Appendix A, the bright mode is directly coupled with the optical mode and can be cooled down by the anti-Stokes interaction. If $G_{1}\neq G_{2}$, the bright mode operator can be written as $Q_{+}=\frac {1}{\sqrt {G_{1}^{2}+G_{2}^{2}}} (G_{1}q_{1}+G_{2}q_{2})$, and $Q_{+}$ is dominated by the mechanical oscillator with larger coupling strength.

 

Fig. 4. The steady-state average phonon numbers of two oscillators versus $ \kappa / \omega _{1}$ for (a) and $G_{2}/G_{1}$ for (b). For (a), $G_{1}=G_{2}=0.03 \omega _{1}$. For (b), $ \kappa =0.1 \omega _{1}$, $G_{1}=0.03 \omega _{1}$. The other parameters are the same as Fig. 3.

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5. Simultaneous cooling of multiple mechanical oscillators

Without loss of generality, we extend this proposed scheme to a multiple oscillators system. All mechanical oscillators are coupled to the same optical cavity, and the frequencies of the oscillators $j$th ($j=2, \ldots N$) are adjustable. The model is schematically shown in Fig. 5(a). Analog to the calculation of two oscillators, the linearized Langevin equations of multiple mechanical oscillators can be written as $\dot {u} (t)=A(t)u(t)+N(t)$, where the optical mode is also expressed by the quadrature operators. The fluctuation operator vector is $u(t)=\{X_{a}(t),Y_{a}(t),q_{1}(t),$ $p_{1}(t), \ldots q_{N}(t),p_{N}(t)\}$, and the noise operator vector is $N(t)=\{\sqrt {\kappa }X_{a_{in}}(t),\sqrt {\kappa } Y_{a_{in}}(t),$ $0,\xi _{1}(t),\ldots 0,\xi _{N}(t)\}$. The coefficient matrix of Langevin equations about multiple oscillators system can be expressed as

 

Fig. 5. (a) The schematic diagram of the system, where multiple mechanical oscillators simultaneously couple the optical field, and the frequencies of oscillators are modulated. (b) The effective cooling channel, where $Q_{1}$ is the bright mode while $Q_{2}\rightarrow Q_{N}$ are the dark modes.

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In order to obtain the Lyapunov control frequencies, we also apply the dynamic evolution equation of the covariance matrix $\dot {M} (t)=A(t)M(t)+M(t)A^{T}(t)+D$, where $D={\rm {Diag}}[\kappa /2,\kappa /2,0,\gamma _{1}(\bar {n}_{1}+\frac {1}{2}),\ldots 0,\gamma _{N}(\bar {n}_{N}+\frac {1}{2})]$. In the derivation of the average phonon number derivatives of two mechanical oscillators, the terms of the correlation between the optical field and the mechanical oscillator and the thermal fluctuations of mechanical oscillators are neglected, and the approximation has also been applied in a multiple oscillators system. Therefore, the equations of the average phonon number derivatives can be expressed as

Here, we consider the cooling effect in the case of the triple oscillators. According to Eq. (17), by calculating the covariance matrix and applying Eq. (18), the dynamic of Lyapunov functions of the triple oscillators system are

Figure 6(a) plots the dynamic time evolution of the average phonon number, and we can see that all the mechanical oscillators can be cooled down to the ground state when the system tends to be stable. Here, $\omega _{1}=\omega _{2}=\omega _{3}$, and the initial values satisfy $\omega _{r_{2}}(0)=\omega _{r_{3}}(0)=\omega _{1}$. After the frequencies modulation, the frequencies of oscillators 2 and 3 will return to the initial eigenfrequencies in the steady state, shown in Fig. 6 (b). By introducing new momentum quadratures and position quadratures to illustrate this cooling mechanism, the corresponding relation can be written as $\mathbf {U}=T_{tr} \mathbf {V}$, where $\mathbf {U}=(O_{1},O_{2},O_{3})^{T}$, $O=Q(P)$, and $\mathbf {V}=(o_{1},o_{2},o_{3})^{T}$, $o=q(p)$. The matrix $T_{tr}$ is composed of an orthonormal basis in vector space $\mathbb {R}^{3}$, which can be expressed as $T_{tr}=(\alpha _{1},\alpha _{2},\alpha _{3})^{T}$ and $\alpha _{1}=\frac {1}{\sqrt {3}}(1,1,1)$, $\alpha _{2}=\frac {1}{\sqrt {2}} (-1,1,0)$, $\alpha _{3}=\sqrt {\frac {2}{3}}(-\frac {1}{2},-\frac {1}{2},1)$. Here, the effective coupling strength of the three mechanical oscillators is the same. The transformed Hamiltonian is reexpressed as

 

Fig. 6. (a) The time evolution of the average phonon numbers for three oscillators. (b) The time-varying control fields for oscillators 2 and 3. The parameters are $\bar {n}_{1}=\bar {n}_{2}=\bar {n}_{3}=50$, $ \kappa =0.1 \omega _{1}$, $ \gamma _{1}= \gamma _{2}= \gamma _{3}=10^{-6} \omega _{1}$, $G_{1}=G_{2}=G_{3}=0.038 \omega _{1}$, $c_{1}=0.66$, $c_{2}=0.61$, $\Delta _{c}=0.99 \omega _{1}$, $ \omega _{1}= \omega _{2}= \omega _{3}$.

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6. Discussion and conclusion

We now discuss the feasibility of the experiment. The mechanical resonator with tunable frequency can be realized experimentally [48] and this mechanism has been widely applied to the quantum level. For instance, Luo et al. [49] have proposed the experimental observation of strong indirect coupling between separated mechanical resonators by controlling the resonant frequency of the phonon cavity. Zhang [50] et al. have reported tunable coherent phonon dynamic with an architecture comprising three graphene mechanical resonators coupled in series, and the frequencies of the resonators are tuned by dc voltages on the gates individually. In ref. [51] the cooling can be experimentally achieved, where $\omega _{1,2}=2\pi \times 9.2~{{\rm {MHz}}}$, $\kappa =2\pi \times 678~{{\rm {KHz}}}$, $\gamma _{1}=2\pi \times 105\pm 8~{{\rm {Hz}}}$ and $\gamma _{2}=2\pi \times 71\pm 19~{{\rm {Hz}}}$, so it means that $\kappa /\omega _{1,2}\sim 0.073$, $\gamma /\omega _{1,2}\sim 7.7\times 10^{-6}$. In the current scheme, the parameters are $\kappa /\omega _{1,2}=0.1$, $\gamma /\omega _{1,2}=10^{-6}$, which are reachable according the experiment [51]. Furthermore, the effective optomechanical coupling is equivalent to a product of the bare one and the cavity density. The cavity density can be modulated by classical driving and it had been achieved experimentally large intracavity photon capacities $10^{5}$ [52], so the final optomechanical coupling can be adjusted suitably.

In this paper, we propose a scheme to achieve the simultaneous ground-state cooling of identical mechanical oscillators by an open-loop Lyapunov control. Due to the tunable frequencies, the effective coupling channel between the bright and dark modes is constructed so that the simultaneous ground-state cooling of identical mechanical oscillators can be achieved. First, we analyze the simultaneous ground-state cooling of two mechanical oscillators and explain the reason for the generation of effective coupling between bright and dark modes in Appendix A. We also numerically plot the mean phonon number in the steady state varying with the cavity dissipation and the ratio of the effective optomechanical coupling. Without loss of generality, we extend this cooling scheme to a multiple oscillators system. With the increase of the coupled mechanical oscillators, the effective couplings between the bright mode and the dark modes can assist in the simultaneous cooling of identical oscillators. In Appendix B, we illuminate the reason in detail and give a generalized expression of the effective coupling.

Appendix A Derivation of effective coupling

In order to illustrate the generation of effective coupling between the bright and dark modes through the tunable frequency, we redefine the new operators, ($Q_{+}$, $P_{+}$) (bright mode) and ($Q_{-}$, $P_{-}$) (dark mode). Here, we make $\omega _{1}=\omega _{2}$ and $G_{1}=G_{2}$, so the new momentum quadratures and position quadratures can be defined as

(22)$$\begin{aligned} Q_{+}&=\frac{1}{\sqrt{2}}(q_{1}+q_{2}),~~~P_{+}=\frac{1}{\sqrt{2}}(p_{1}+p_{2}), \\ Q_{-}&=\frac{1}{\sqrt{2}}(q_{1}-q_{2}),~~~P_{-}=\frac{1}{\sqrt{2}}(p_{1}-p_{2}). \end{aligned}$$

After the conversion of operators, the commutations relations and the conservation of energy should be satisfied. Applying Eq. (23), the transformed Hamiltonian can be written as

(23)$$\begin{aligned} H&=\Delta_{c}a^{\dagger}a+\frac{\omega_{1}}{2}P_{+}^{2}+\frac{\omega_{1}}{2}P_{-}^{2} +(\frac{\omega_{1}}{4}+\frac{\omega_{r}^{2}(t)}{4\omega_{1}})Q_{+}^{2} +(\frac{\omega_{1}}{4}+\frac{\omega_{r}^{2}(t)}{4\omega_{1}})Q_{-}^{2} \\ & +\sqrt{2}G_{1}(a+a^{\dagger})Q_{+} +(\frac{\omega_{1}}{2}-\frac{\omega_{r}^{2}(t)}{2\omega_{1}})Q_{+}Q_{-}. \end{aligned}$$

From Eq. (24), we can see that if $\omega _{r}(t)=\omega _{1}$, the last term will disappear, the dark mode decouples with the system, and the two oscillators can not be cooled simultaneously because the two oscillators can be in dark mode with large mean values, the so called dark mode effect. By introducing the Lyapunov control, an effective coupling between $Q_{+}$ and $Q_{-}$ is formed so that there is a cascade cooling channel, as shown in Fig. 7(a). In Fig. 7(b), we plot the time dependence of the coupling between bright mode and dark mode, which is written as $g(t)=\frac {\omega _{1}}{2}-\frac {\omega _{r}^{2}(t)}{2\omega _{1}}$. It is obviously seen that $g(t)=0$ at the beginning because the control field has not been applied yet. After the frequency modulation, the mechanical oscillator 2 will return the eigenfrequency so the coupling vanishes again.

 

Fig. 7. (a) The effective cooling channel, where $Q_{+}$ is the bright mode while $Q_{-}$ is the dark mode. (b) The time evolution of the effective coupling between bright mode and dark mode. Here, the parameters are the same as Fig. 2.

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Appendix B The generalized effective coupling

Here, we deduce the generalized effective coupling between the bright and the dark modes of the identical mechanical oscillators. The Gram-Schmidt method is applied to define an orthonormal basis that can obviously express the coupling between the bright and dark modes. A generalized momentum quadrature and position quadrature representing the bright mode can be defined as $\mathcal {P}_{1}=1/\sqrt {N}\sum _{n}p_{n}$ and $\mathcal {Q}_{1}=1/\sqrt {N}\sum _{n}q_{n}$, while dark modes can be defined as $\mathcal {P}_{k}=\sum _{n}\alpha _{kn}p_{n}$ as well as $\mathcal {Q}_{k}=\sum _{n}\alpha _{kn}q_{n}$ and commutations are satisfied. The coefficients need to satisfy $\sum _{n}\alpha _{nk}=0$ and $\sum _{n}\alpha _{nk}^{*}\alpha _{nk^{\prime }}=\delta _{kk^{\prime }}$.

The compact form of the Langevin equations which are redefined by the operators ($\mathcal {P}_{k},\mathcal {Q}_{k}$) is

(24)$$\dot{\textbf{u}}=TA(t)T^{\top}\textbf{u}+\textbf{u}_{in},$$

where $\textbf {u}=T\textbf {v}$ and the vectors $\textbf {u}=(X_{a},Y_{a},\mathcal {Q}_{1}, \mathcal {P}_{1},\ldots \mathcal {Q}_{N},\mathcal {P}_{N})$, $\textbf {v}=(X_{a},Y_{a},q_{1},p_{1}, \ldots q_{N},p_{N})$. $\textbf {u}_{in}$ is the total noise term. The coefficient matrix A(t) is the same as Eq. (16), while in the following derivation, all optomechanical coupling strengths, dissipations, and frequencies of the mechanical oscillators are the same which are equal to $G$, $\gamma$, and $\omega$. The transfer matrix $T$ is defined as

(25)$$\begin{gathered}T= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & \alpha_{11} & 0 & \alpha_{12} & 0 & \cdots & \alpha_{1N} & 0\\ 0 & 0 & 0 & \alpha_{11} & 0 & \alpha_{12} & \cdots & 0 & \alpha_{1N}\\ 0 & 0 & \alpha_{21} & 0 & \alpha_{22} & 0 & \cdots & \alpha_{2N} & 0\\ 0 & 0 & 0 & \alpha_{21} & 0 & \alpha_{22} & \cdots & 0 & \alpha_{2N}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \alpha_{N1} & 0 & \alpha_{N2} & 0 & \cdots & \alpha_{NN} & 0\\ 0 & 0 & 0 & \alpha_{N1} & 0 & \alpha_{N2} & \cdots & 0 & \alpha_{NN}\\ \end{pmatrix}, \end{gathered}$$

where the vector $(\alpha _{11}, \alpha _{12}, \ldots, \alpha _{1N})=1/\sqrt {N}(1,1,\ldots,1)$ represents the bright mode. Therefore, the new Langevin equations of the bright mode $(\mathcal {Q}_{1},\mathcal {P}_{1})$ can be written as

(26)$$\begin{aligned} \dot{\mathcal{Q}}_{1}&=\sum_{k^{\prime}=1}^{N} g_{(1,k^{\prime})}\mathcal{P}_{k^{\prime}}, \\ \dot{\mathcal{P}}_{1}&={-}\sum_{k^{\prime}=1}^{N}g_{(1,k^{\prime})}(t)\mathcal{Q}_{k^{\prime}} +\sqrt{2N}GX_{a}-\frac{\gamma}{2}\mathcal{P}_{1} +\Xi_{1}, \end{aligned}$$

and the associated coefficients are defined as

(27)$$\begin{aligned}g_{(1,k^{\prime})}&= \sum_{n=1}^{N}\alpha_{(1,n)}\alpha_{(k^{\prime},n)}\omega, \\ g_{(1,k^{\prime})}(t)&= \frac{\omega}{\sqrt{N}}\alpha_{(k^{\prime},1)}+ \sum_{n=1}^{N-1}\alpha_{(1,n+1)}\alpha_{(k^{\prime},n+1)} \frac{\omega_{r_{n+1}}^{2}(t)}{\omega}, \end{aligned}$$

where the effective coupling between the bright and dark modes is $g_{(1,k^{\prime })}(t)$, ($k^{'}=2,3, \ldots N$). $\Xi _{1}$ is the noise term. From Eq. (26) we can see that the coupling $g_{(1,k^{\prime })}(t)$ between the bright and dark modes is reconstructed by the Lyapunov control, and the schematic diagram of the cooling channel is shown in Fig. 5(b). By modulating the tunable frequencies, the applicable cooling channels are formed so that the simultaneous cooling of multiple identical mechanical oscillators can be achieved.

Funding

National Key Research and Development Program of China (2021YFE0193500); National Natural Science Foundation of China (11874099).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]  

2. Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018). [CrossRef]  

3. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “$\mathcal {PT}$-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014). [CrossRef]  

4. C. A. Regal and K. W. Lehnert, “From cavity electromechanics to cavity optomechanics,” J. Phys.: Conf. Ser. 264, 012025 (2011). [CrossRef]  

5. H.-K. Lau and A. A. Clerk, “Ground-state cooling and high-fidelity quantum transduction via parametrically driven bad-cavity optomechanics,” Phys. Rev. Lett. 124(10), 103602 (2020). [CrossRef]  

6. Y. Wei, X. Wang, B. Xiong, C. Zhao, J. Liu, and C. Shan, “Improving few-photon optomechanical effects with coherent feedback,” Opt. Express 29(22), 35299–35313 (2021). [CrossRef]  

7. H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016). [CrossRef]  

8. J.-Q. Liao, Q.-Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014). [CrossRef]  

9. Y. Guo, K. Li, W. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014). [CrossRef]  

10. E. E. Wollman, C. Lei, A. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. Schwab, “Quantum squeezing of motion in a mechanical resonator,” Science 349(6251), 952–955 (2015). [CrossRef]  

11. F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Quantum nondemolition measurement of a nonclassical state of a massive object,” Phys. Rev. X 5(4), 041037 (2015). [CrossRef]  

12. C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556(7702), 478–482 (2018). [CrossRef]  

13. A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10(9), 095010 (2008). [CrossRef]  

14. J. Suh, A. Weinstein, C. Lei, E. Wollman, S. Steinke, P. Meystre, A. A. Clerk, and K. Schwab, “Mechanically detecting and avoiding the quantum fluctuations of a microwave field,” Science 344(6189), 1262–1265 (2014). [CrossRef]  

15. I. Shomroni, L. Qiu, D. Malz, A. Nunnenkamp, and T. J. Kippenberg, “Optical backaction-evading measurement of a mechanical oscillator,” Nat. Commun. 10(1), 2086 (2019). [CrossRef]  

16. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99(9), 093902 (2007). [CrossRef]  

17. M. Asjad, N. E. Abari, S. Zippilli, and D. Vitali, “Optomechanical cooling with intracavity squeezed light,” Opt. Express 27(22), 32427–32444 (2019). [CrossRef]  

18. X. Xu, T. Purdy, and J. M. Taylor, “Cooling a harmonic oscillator by optomechanical modification of its bath,” Phys. Rev. Lett. 118(22), 223602 (2017). [CrossRef]  

19. Y.-L. Liu and Y.-X. Liu, “Energy-localization-enhanced ground-state cooling of a mechanical resonator from room temperature in optomechanics using a gain cavity,” Phys. Rev. A 96(2), 023812 (2017). [CrossRef]  

20. J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109(10), 103603 (2012). [CrossRef]  

21. U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, and M. Aspelmeyer, “Cooling of a levitated nanoparticle to the motional quantum ground state,” Science 367(6480), 892–895 (2020). [CrossRef]  

22. S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108(15), 153601 (2012). [CrossRef]  

23. L. Qiu, I. Shomroni, P. Seidler, and T. J. Kippenberg, “Laser cooling of a nanomechanical oscillator to its zero-point energy,” Phys. Rev. Lett. 124(17), 173601 (2020). [CrossRef]  

24. C. Sommer and C. Genes, “Partial optomechanical refrigeration via multimode cold-damping feedback,” Phys. Rev. Lett. 123(20), 203605 (2019). [CrossRef]  

25. D.-G. Lai, J. Huang, B.-P. Hou, F. Nori, and J.-Q. Liao, “Domino cooling of a coupled mechanical-resonator chain via cold-damping feedback,” Phys. Rev. A 103, 063509 (2021). [CrossRef]  

26. C.-H. Bai, D.-Y. Wang, S. Zhang, S. Liu, and H.-F. Wang, “Double-mechanical-oscillator cooling by breaking the restrictions of quantum backaction and frequency ratio via dynamical modulation,” Phys. Rev. A 103(3), 033508 (2021). [CrossRef]  

27. C. Genes, D. Vitali, and P. Tombesi, “Simultaneous cooling and entanglement of mechanical modes of a micromirror in an optical cavity,” New J. Phys. 10(9), 095009 (2008). [CrossRef]  

28. A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D. Kashkanova, C. Deutsch, J. Reichel, and J. G. E. Harris, “Optically mediated hybridization between two mechanical modes,” Phys. Rev. Lett. 112(1), 013602 (2014). [CrossRef]  

29. F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3(1), 987 (2012). [CrossRef]  

30. M. T. Naseem and Ö. E. Müstecaplıoğlu, “Ground-state cooling of mechanical resonators by quantum reservoir engineering,” Commun. Phys. 4(1), 95 (2021). [CrossRef]  

31. D.-G. Lai, J.-F. Huang, X.-L. Yin, B.-P. Hou, W. Li, D. Vitali, F. Nori, and J.-Q. Liao, “Nonreciprocal ground-state cooling of multiple mechanical resonators,” Phys. Rev. A 102(1), 011502 (2020). [CrossRef]  

32. C. Sommer, A. Ghosh, and C. Genes, “Multimode cold-damping optomechanics with delayed feedback,” Phys. Rev. Res. 2(3), 033299 (2020). [CrossRef]  

33. Z. Yang, C. Zhao, R. Peng, S.-L. Chao, J. Yang, and L. Zhou, “The simultaneous ground-state cooling and synchronization of two mechanical oscillators by driving nonlinear medium,” Annalen der Physik 534(5), 2100494 (2022). [CrossRef]  

34. M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J.-P. Wolf, and H. Rabitz, “Quantum control of tightly competitive product channels,” Phys. Rev. Lett. 102(25), 253001 (2009). [CrossRef]  

35. F. Delgado, “Quantum control on entangled bipartite qubits,” Phys. Rev. A 81(4), 042317 (2010). [CrossRef]  

36. C. Li, J. Song, Y. Xia, and W. Ding, “Driving many distant atoms into high-fidelity steady state entanglement via lyapunov control,” Opt. Express 26(2), 951–962 (2018). [CrossRef]  

37. K. De Greve, P. L. McMahon, D. Press, T. D. Ladd, D. Bisping, C. Schneider, M. Kamp, L. Worschech, S. Höfling, A. Forchel, and Y. Yamamoto, “Ultrafast coherent control and suppressed nuclear feedback of a single quantum dot hole qubit,” Nat. Phys. 7(11), 872–878 (2011). [CrossRef]  

38. Z. C. Shi, X. L. Zhao, and X. X. Yi, “Robust state transfer with high fidelity in spin-1/2 chains by lyapunov control,” Phys. Rev. A 91(3), 032301 (2015). [CrossRef]  

39. D. Ran, W.-J. Shan, Z.-C. Shi, Z.-B. Yang, J. Song, and Y. Xia, “Generation of nonclassical states in nonlinear oscillators via lyapunov control,” Phys. Rev. A 102(2), 022603 (2020). [CrossRef]  

40. W. Li, C. Li, and H. Song, “Quantum synchronization in an optomechanical system based on lyapunov control,” Phys. Rev. E 93(6), 062221 (2016). [CrossRef]  

41. B. Xiong, X. Li, S.-L. Chao, Z. Yang, W.-Z. Zhang, W. Zhang, and L. Zhou, “Strong mechanical squeezing in an optomechanical system based on lyapunov control,” Photonics Res. 8(2), 151–159 (2020). [CrossRef]  

42. S. C. Hou and X. X. Yi, “Quantum lyapunov control with machine learning,” Quantum Info. Process. 19(1), 1–20 (2020). [CrossRef]  

43. S. C. Hou, M. A. Khan, X. X. Yi, D. Dong, and I. R. Petersen, “Optimal lyapunov-based quantum control for quantum systems,” Phys. Rev. A 86(2), 022321 (2012). [CrossRef]  

44. R. A. Barton, I. R. Storch, V. P. Adiga, R. Sakakibara, B. R. Cipriany, B. Ilic, S. P. Wang, P. Ong, P. L. McEuen, J. M. Parpia, and H. G. Craighead, “Photothermal self-oscillation and laser cooling of graphene optomechanical systems,” Nano Lett. 12(9), 4681–4686 (2012). [CrossRef]  

45. P. Weber, J. Güttinger, I. Tsioutsios, D. E. Chang, and A. Bachtold, “Coupling graphene mechanical resonators to superconducting microwave cavities,” Nano Lett. 14(5), 2854–2860 (2014). [CrossRef]  

46. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77(3), 033804 (2008). [CrossRef]  

47. D.-G. Lai, F. Zou, B.-P. Hou, Y.-F. Xiao, and J.-Q. Liao, “Simultaneous cooling of coupled mechanical resonators in cavity optomechanics,” Phys. Rev. A 98(2), 023860 (2018). [CrossRef]  

48. C. Chen, S. Lee, V. V. Deshpande, G.-H. Lee, M. Lekas, K. Shepard, and J. Hone, “Graphene mechanical oscillators with tunable frequency,” Nat. Nanotechnol. 8(12), 923–927 (2013). [CrossRef]  

49. G. Luo, Z.-Z. Zhang, G.-W. Deng, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, L. Tian, and G.-P. Guo, “Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity,” Nat. Commun. 9(1), 1–6 (2018). [CrossRef]  

50. Z.-Z. Zhang, X.-X. Song, G. Luo, Z.-J. Su, K.-L. Wang, G. Cao, H.-O. Li, M. Xiao, G.-C. Guo, L. Tian, G.-W. Deng, and G.-P. Guo, “Coherent phonon dynamics in spatially separated graphene mechanical resonators,” Proc. Natl. Acad. Sci. U. S. A. 117(11), 5582–5587 (2020). [CrossRef]  

51. C. F. Ockeloen-Korppi, M. F. Gely, E. Damskägg, M. Jenkins, G. A. Steele, and M. A. Sillanpää, “Sideband cooling of nearly degenerate micromechanical oscillators in a multimode optomechanical system,” Phys. Rev. A 99(2), 023826 (2019). [CrossRef]  

52. M. J. Burek, J. D. Cohen, S. M. Meenehan, N. El-Sawah, C. Chia, T. Ruelle, S. Meesala, J. Rochman, H. A. Atikian, M. Markham, D. J. Twitchen, M. D. Lukin, O. Painter, and M. Lončar, “Diamond optomechanical crystals,” Optica 3(12), 1404–1411 (2016). [CrossRef]