The Evolution of Squeezing in Coupled Macroscopic Mechanical Oscillator Systems

Abstract

Quantum squeezing in macroscopic oscillator systems plays a critical role in bridging quantum mechanics with classical-scale phenomena, enabling high-precision measurements and fundamental tests of quantum physics. In this work, we investigate the effect of squeezing on the phonon state in a hybrid macroscopic mechanical system consisting of an ensemble of Rydberg atoms coupled to two macroscopic mechanical oscillators. We notice that the dipole–dipole coupling between atoms and mechanical oscillators can be transferred to the indirectly coupled mechanical interaction, and the nonlinear effective Hamiltonian can be solved to generate a squeezed effect on the mechanical mode. We also discuss the noise effects induced by amplitude and phase fluctuations on the squeezed quadratures of the system.

1. Introduction

Quantum squeezing in macroscopic oscillator systems plays a critical role in bridging quantum mechanics with classical-scale phenomena, enabling high-precision measurements and fundamental tests of quantum physics [1,2]. Recent research has made significant progress in the study of quantum squeezing effects and their applications in solid-state quantum systems [3] and optomechanical systems [4,5]. Meanwhile, we noticed that the solid-state hybrid systems are playing important roles in the realization of quantum information processing (QIP). For example, the realization of light–matter interaction between microresonators coupled with various quantum bits (qubits) has attracted much attention in the past few decades. And these separate qubits can be used as the quantum nodes in quantum information processing and quantum networks [6,7,8,9]. For further applications, multi-qubit quantum logic gate operations have been realized, such as quantum controlled-not gate operations based on trapped ions [10,11], semiconductor quantum dots [12,13], and also photonic qubits [14,15,16]. Meanwhile, the characterization and quantification of both squeezing phenomena and quantum entanglement have emerged as important research focuses, given their fundamental significance in quantum optics and quantum information science [17,18,19,20].

Cavity quantum electrodynamics (Cavity-QED) studies the strong interaction between confined optical fields and quantum emitters, such as dipole defects in microresonators. This system provides an ideal platform for quantum information processing (QIP), enabling efficient photon–solid-state qubit interactions. During the past few decades, numerous different systems have been used as the Cavity-QED system, such as a single Rydberg atom coupled with a Fabry–Pérot microwave cavity, nitrogen-vacancy (NV) centers coupled to an optical cavity, a superconducting qubit coupled with a microwave cavity, and so on. For example, in [21], the authors illustrated the dynamics of a single atom coupled with a microcavity resonator and indicated that the single atom within the resonator can dynamically control the output field of the cavity. Using the NV-centers coupled to photonic cavities, the resonant zero phonon line (ZPL) relevant to the emitted photons from nitrogen-vacancy centers can be significantly enhanced [22,23], and the quantum non-demolition (QND) measurement on the electronic spin state was also sketched [24,25].

Furthermore, based on the Fabry–Pérot cavity system, we can induce the mechanical oscillation by using a flexible mirror that oscillates around an equilibrium position on one side of the Fabry–Pérot cavity. This can be treated as a harmonic oscillation when the flexible mirror is operated at its resonance frequency. This motion modulates the cavity length, generating radiation pressure on the mirror. Such dynamics establish an optomechanical coupling between the cavity field and the oscillating mirror. Both theoretical and experimental studies have extensively explored this coherent interaction, demonstrating strong coupling between a quantized mechanical resonator, the optical field, and dipole defects [26,27,28,29,30].

As discussed in Ref. [31], the authors proposed a scheme to control macroscopic mechanical oscillators using a single atom. It is shown that the dipole–dipole coupling between the atomic Rydberg transition and the mechanical resonators allows cooling the oscillators. And the mechanical entanglement could be generated on the phonon state. Here in this paper, we focus on the quantum optomechanics between the macroscopic mechanical oscillators and Rydberg atoms, and investigate the interaction between the mechanical oscillators through the phonon exchange process in an atom-mechanical oscillators hybrid system theoretically. A squeezed state could be generated on the mechanical mode by classical driving, and the noise effect induced by phase and amplitude fluctuations is further discussed. Quantum squeezing represents a highly promising non-classical phenomenon, with sustained research efforts in recent years yielding significant breakthroughs across multiple platforms. These include the generation of squeezed states in circuit quantum electrodynamical systems [32], superconducting transmission line resonators [33,34], spin condensates [35], cavity optomechanical systems [36,37], quantum devices [38], and quantum-enhanced computing architectures [39]. Given its fundamental importance, squeezing effects are anticipated to play increasingly vital roles in quantum information processing and quantum control applications.

In this work, we investigate phonon-state squeezing in a hybrid macroscopic quantum system comprising an ensemble of Rydberg atoms coupled to mechanical cantilevers. We demonstrate that the interactions between atoms and oscillations can be effectively transferred to mediate indirect mechanical coupling. Through analytical solution of the nonlinear effective Hamiltonian, we show that this coupling generates significant squeezing in the mechanical mode. Furthermore, we characterize the influence of amplitude and phase noise on the squeezed quadratures, revealing their impact on the system’s quantum coherence.

2. The Models

The schematic diagram of the system is shown in Figure 1. We study a system composed of a Rydberg atom ensemble coupled with two electrically charged cantilevers, where the distance of the cantilevers’ oscillation is denoted as d, and the two cantilevers are charged with electric charges $\pm Q$ on opposite tips.

Assuming that the atoms are prepared in the ground energy state, the total Hamiltonian of the system is described as

$$\begin{array}{c}\hfill H=H_{can}+H_{at}+V_{int},\end{array}$$

(1)

Here, $H_{can}=\hslash {\omega }_a{a}^{†}a+\hslash {\omega }_b{b}^{†}b$ represents the Hamiltonian of the cantilevers, a and b are the annihilation operators of the fundamental mode of the mechanical oscillators, respectively. $H_{at}=-\hslash /2{\displaystyle \sum _{i=1,N}}{\Omega }_i{\sigma }_{z,i}$ denotes the Hamiltonian of the Rydberg atoms in the ensemble, and ${\sigma }_{z,i}=\left(\right|s\rangle \langle s|-{|e\rangle \langle e\left|\right)}_i$. Here, $\Omega$ represents the eigenfrequencies of the atoms. $|s\rangle$ and $|e\rangle$ represent the metastable state and the excited state, respectively. $V_{int}$ describes the interaction between the atoms and the cantilevers. The characteristic parameters of the cantilevers include the effective mass $m_{eff}$, the length l, and the zero-point motion of the resonator mode $x_{zp}=\sqrt{\hslash /2m_{eff}\omega }$ with frequency $\omega$. The dynamics of the evolution could be described as follows: the atoms are initially prepared in their ground state. When a single Rydberg excitation is introduced, the resulting energy shift brings neighboring atoms out of resonance with the excitation laser. This effect, known as the Rydberg blockade, suppresses multiple excitations within the blockade radius, effectively limiting the system to a single Rydberg excitation. The blockade mechanism enables an enhanced photon–atom interaction cross-section while simultaneously creating an ideal interface between individual atoms and mechanical resonators. The interaction Hamiltonian $H_{int}$ can be decomposed into the electric dipole term $V_{dipole}$ and the mechanical oscillator coupling term $V_{couple}$; the first term of the interaction Hamiltonian could be represented as $V_{dipole}=-{\displaystyle \sum _l}{E}_l·{\mu }_{at}$, where $E_l$ is the electric field, and ${\mu }_{at}$ denotes the dipole element of the atomic-level transition.

Assuming that the frequency of the oscillator is resonant with the Rydberg transition $|s\rangle \to |e\rangle$, the Hamiltonian of the atom–mechanical oscillator interaction term can be reduced to $V_{couple}=\hslash G_a(a+a^{†})({\sigma }_{+}+{\sigma }_{-})+\hslash G_b(b+b^{†})({\sigma }_{+}+{\sigma }_{-})$, where $G_{a\left(b\right)}=Q{x}_{zp,a\left(b\right)}\mu /(4\hslash \pi {ϵ}_0{R}^3)$ is the coupling strength, and R represents the distance between the atomic ensembles and the cantilevers. Q denotes the pure electric charges on the two cantilevers. The atomic level transition operators are ${\sigma }_{+}=|e\rangle \langle s|$ and ${\sigma }_{-}=|s\rangle \langle e|$. Under the rotating-wave approximation (RWA), the interaction Hamiltonian $V_{couple}$ can be reduced to

$$\begin{array}{c}\hfill V=\hslash G_a(a{\sigma }_{+}+a^{†}{\sigma }_{-})+\hslash G_b(b{\sigma }_{+}+b^{†}{\sigma }_{-}).\end{array}$$

(2)

As discussed in Ref. [31], $V_{couple}$ describes the phonon exchange interaction between the cantilevers, which is negligible compared with the atom–cantilever interaction. So, it is hard to exploit the direct interaction between the two mechanical oscillators.

The total Hamiltonian of the atom–mechanical oscillator hybrid system can be expressed as

$$\begin{array}{c}\hfill H=\hslash {\omega }_a{a}^{†}a+\hslash {\omega }_b{b}^{†}b-{\displaystyle \frac{1}{2}}\hslash \Omega {\sigma }_z+\hslash G_a(a{\sigma }_{+}+a^{†}{\sigma }_{-})+\hslash G_b(b{\sigma }_{+}+b^{†}{\sigma }_{-}),\end{array}$$

(3)

where $G_a$ and $G_b$ are the coupling coefficients between the first and second cantilevers and the atoms. Here, we assume the detunings between the atom and two cantilevers satisfy the large detuning limit, that is, $|{\Delta }_{a\left(b\right)}|=|\Omega -{\omega }_{a\left(b\right)}|\gg \sqrt{G_a^2+G_b^2}$. The mode frequency of the mechanical oscillator relies on the dimensions of the cantilever $(l,t)$, the density $\rho$, and the Young’s modulus E as $\omega /2\pi =3.516(t/l^2)\sqrt{E/\left(12\rho \right)}$. By adjusting the dimension parameters l and t, the large detuning could be fulfilled beforehand.

The strong interaction of the system depends on the charge amount of the cantilevers. Initially, if we increase the charge distribution on one of the cantilevers, say $Q_b$, the dipole–dipole interaction between the atom ensemble and the cantilever b is increased. The increased interaction will introduce the second-order nonlinear interaction on the Hamiltonian [40,41], and the total Hamiltonian, considering the second-order interaction, can be described as

$$\begin{array}{ccc}\hfill H& =& \hslash {\omega }_a{a}^{†}a+\hslash {\omega }_b{b}^{†}b-{\displaystyle \frac{1}{2}}\hslash \Omega {\sigma }_z+\hslash G_a(a+a^{†})({\sigma }_{+}+{\sigma }_{-})\hfill \\ & & +\hslash G_b(b+b^{†})({\sigma }_{+}+{\sigma }_{-})+\hslash \widetilde{G_b}{(b+b^{†})}^2({\sigma }_{+}+{\sigma }_{-}).\hfill \end{array}$$

(4)

Here, $\widetilde{G_b}$ denotes the second-order interaction between the atom and the cantilever. And the interaction between the atom and the cantilever consists of both the first-order interaction part and the second-order interaction part. In the free field case, the operators a and b evolve as follows: $a\left(t\right)=a\left(0\right)e^{-i{\omega }_{a}t}$, $b\left(t\right)=b\left(0\right)e^{-i{\omega }_{b}t}$. For the free atomic case, the energy operators evolve as follows: ${\sigma }_{\pm }\left(t\right)={\sigma }_{\pm }\left(0\right)e^{\pm i\Omega t}$. We choose the resonant conditions that ${\omega }_a=2{\omega }_b$, under the rotating-wave approximation; thus, the Hamiltonian of the system could be described as

$$\begin{array}{c}\hfill H^{\prime }=\hslash {\omega }_a{a}^{†}a+\hslash {\omega }_b{b}^{†}b-{\displaystyle \frac{1}{2}}\hslash \Omega {\sigma }_z+\hslash G_a(a{\sigma }_{+}+a^{†}{\sigma }_{-})+\hslash \widetilde{G_b}(b^2{\sigma }_{+}+b^{†2}{\sigma }_{-}).\end{array}$$

(5)

Under the large detuning conditions $|\Omega -{\omega }_a|=|\Omega -2{\omega }_b|\gg \sqrt{G_a^2+{\widetilde{G_b}}^2}$, we can perform the Frohlich–Nakajima transformation. The transform operator $S^{\prime }$ can be described as follows:

$$\begin{array}{c}\hfill S^{\prime }=\hslash {\Delta }_a(A{a}^{†}{\sigma }_{-}-Ba{\sigma }_{+})+\hslash {\Delta }_b(C{b}^{†2}{\sigma }_{-}-D{b}^2{\sigma }_{+}).\end{array}$$

(6)

According to the relations $[S^{\prime },H_0]=H_I$, the coefficients can be solved as $A=-B=G_a/{\Delta }_a$, and $C=-D=\widetilde{G_b}/{\Delta }_b$. And the effective Hamiltonian of the system after transformation can be solved as $H_{eff}=H_0+{\displaystyle \frac{1}{2}}[H_I,S^{\prime }]$. Then, the effective Hamiltonian can be obtained, and it can be described as

$$\begin{array}{ccc}\hfill H^{\prime }& =& \hslash ({\omega }_a-{\displaystyle \frac{G_a^2}{{\Delta }_a}})a^{†}a{\sigma }_z+\hslash ({\omega }_b-{\displaystyle \frac{{\widetilde{G_b}}^2}{{\Delta }_b}})b^{†}b{\sigma }_z+\hslash ({\displaystyle \frac{2g_b^2}{{\delta }_b}}-{\displaystyle \frac{g_b^2}{{\delta }_b}}{\sigma }_z)\hfill \\ & & -\hslash G_a\widetilde{G_b}({\displaystyle \frac{1}{2{\Delta }_a}}+{\displaystyle \frac{1}{2{\Delta }_b}})(a^{†}{b}^2+a{b}^{†2}){\sigma }_z.\hfill \end{array}$$

(7)

In the interaction picture, if the Rydberg atoms are prepared in the ground state, the nonlinear interaction Hamiltonian under the large detuning conditions can be expressed as

$$\begin{array}{c}\hfill H_I^{\prime }=-{\displaystyle \frac{\hslash G_a\widetilde{G_b}({\Delta }_a+{\Delta }_b)}{2{\Delta }_a{\Delta }_b}}(a^{†}{b}^2{e}^{i\delta t}+a{b}^{†2}{e}^{-i\delta t}).\end{array}$$

(8)

Under the resonant conditions that $\delta =2{\omega }_b-{\omega }_a=0$, we have $H_I^{\prime }=\hslash g(a^{†}{b}^2+a{b}^{†2})$, where $\tilde{g}=\hslash G_a\widetilde{G_b}({\Delta }_a+{\Delta }_b)/\left(2{\Delta }_a{\Delta }_b\right)$ denotes the nonlinear coupling strength between the two cantilevers.

3. Results

Here, we consider the case that one of the cantilevers is driven by a classical field with amplitude $\xi$ and phase $\varphi$, the Hamiltonian could be reduced to

$$\begin{array}{c}\hfill H_I^{\prime }=\hslash g\xi (b^2{e}^{-i\varphi }+b^{†2}{e}^{i\varphi })\end{array}$$

(9)

The operator of the time evolution could be expressed as

$$\begin{array}{c}\hfill S\left(\chi \right)=exp[-ig\xi (b^2{e}^{-i\varphi }+b^{†2}{e}^{i\varphi })]\end{array}$$

(10)

which acts on the mode b of the cantilever; the squeezed state could be generated on the mechanical mode of the cantilever. Here, the squeezed coefficient is described by $\chi =2g\xi t$, in which $g\xi$ represents the Rabi frequency. Suppose that the mechanical mode of the cantilever is cooled in the vacuum state $|0\rangle$; the squeezed state could be generated by the transformation $|\chi \rangle =S(\chi \left)\right|0\rangle$. We define the quadratures of the squeezed state as $X_1=(b+b^{†})/2$ and $X_2=(b-b^{†})/\left(2i\right))$. The variance of the two quadratures can be express as $\Delta X_1^2=\langle X_1^2\rangle -{\langle X_1\rangle }^2=e^{-2\chi }/4$ and $\Delta X_2^2=\langle X_2^2\rangle -{\langle X_2\rangle }^2=e^{2\chi }/4$. It is obvious that the squeezed effect could be tuned macroscopically by changing the charges, the distance between the atomic ensemble and the cantilever, and the mechanical properties of the cantilever. Here, as shown in Figure 2, we numerically studied the quadrature variance under the alteration of the charge and time in ideal conditions.

Meanwhile, the noise will affect the variances of the two quadratures, $X_1$ and $X_2$, as well as the squeezing properties. Here, we focus on the noise effect that relies on the amplitude fluctuation and phase fluctuation introduced by the classical microwave pulse. As both the variance of the amplitude fluctuations and the linewidth of the pulse will affect the final amplitude fluctuation, we can conclude that pumping microwaves are the primary contributors to amplitude fluctuation and phase fluctuation. Here, we denote the variance as $\zeta$ and the bandwidth of the microwave linewidth as $\tau$.

The influence of the amplitude fluctuation on the squeezing efficiency of the mechanical mode can be written explicitly as

$$\begin{array}{c}\hfill {\langle \Delta X_{1\left(2\right)}\rangle }^2={\displaystyle \frac{1}{4}}exp[4It+{\displaystyle \frac{4I}{\tau }}(e^{-\tau t}-1)\mp 2\chi t].\end{array}$$

(11)

Under the noise effect, the phase fluctuation is described by $\delta \varphi \left(t\right)$ with expectation value $\langle \delta \varphi \left(t\right)\rangle =0$. It induces the linewidth D as $\langle \delta \dot{\varphi }\left(t\right)\delta \dot{\varphi }{(}^{\prime })\rangle =2D\delta (t-t^{\prime })$ on the field. By solving the Heisenberg equation under phase fluctuation, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}(b^{†}b+b{b}^{†})=-4g\xi (b^2{e}^{i\delta \varphi \left(t\right)}+b^{†2}{e}^{-i\delta \varphi \left(t\right)}),\end{array}$$

(12)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(b^2{e}^{i\delta \varphi \left(t\right)}\right)=-2g\xi (b^{†}b+b{b}^{†})+i\delta \dot{\varphi }\left(t\right)b^2{e}^{i\delta \varphi \left(t\right)},\end{array}$$

(13)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(b^{†2}{e}^{-i\delta \varphi \left(t\right)}\right)=-2g\xi (b^{†}b+b{b}^{†})-i\delta \dot{\varphi }\left(t\right)b^{†2}{e}^{-i\delta \varphi \left(t\right)}.\end{array}$$

(14)

Then, we can derive the matrix form of the time evolution equation on the basis of $\Psi ={(b^{†}b+b{b}^{†},b^2{e}^{i\delta \varphi \left(t\right)},b^{†2}{e}^{-i\delta \varphi \left(t\right)})}^T$, which could be expressed below

$$\begin{array}{c}\hfill {\displaystyle \frac{d\Psi }{dt}}=K\Psi +i\delta \dot{\varphi }\left(t\right)K_0\Psi .\end{array}$$

(15)

Here, the expressions of K and $K_0$ are $K=\left(\begin{array}{ccc}0& -4g\xi & -4g\xi \\ -2g\xi & 0& 0\\ -4g\xi & 0& 0\end{array}\right)$ and $K_0=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right)$.

Meanwhile, the definition of the time evolution function on the basis of $\Phi ={(b^2,(b^{†}b+b{b}^{†})e^{-i\delta \varphi \left(t\right)},b^2{e}^{i\delta \varphi \left(t\right)})}^T$ can be described as

$$\begin{array}{c}\hfill {\displaystyle \frac{d\Phi }{dt}}=K\Phi +i\delta \dot{\varphi }\left(t\right)K_0\Phi .\end{array}$$

(16)

Suppose that the initial state of the photon is in the vacuum state; the values of $\langle b^2\rangle$, $\langle b^{†2}\rangle$ and $\langle b^{†}b+b{b}^{†}\rangle$ could be solved [42]. Then, the quadrature variance under the phase fluctuation can be described as

$$\begin{array}{ccc}\hfill {\langle \Delta X_{1\left(2\right)}\rangle }^2& =& {\displaystyle \frac{1}{4}}[e^{-2\tau t}{\displaystyle \frac{2\tau g\chi }{2g^2{\chi }^2-{\tau }^2/8}}-e^{(2g\chi -3\tau /2)t}{\displaystyle \frac{2g\chi +5\tau /2}{4g\chi +\tau }}+e^{-(2g\chi +3\tau /2)t}{\displaystyle \frac{2g\chi -5\tau /2}{4g\chi -\tau }}\hfill \\ & & \pm {\displaystyle \frac{e^{-\tau t/2}\tau sinh\left(2g\chi t\right)}{\sqrt{{\tau }^2+16g^2{\chi }^2}}}\pm e^{-\tau t/2}cosh\left(2g\chi t\right)].\hfill \end{array}$$

(17)

Next, we numerically studied the variance of the squeezed state quadratures under the fluctuations, and the results are shown in Figure 3.

In the simulation, the experiment parameters were set as $G_a=1$ MHz which corresponds to $Q_a=3\times {10}^{3}e$ of the charge distribution. The second-order coupling constant of the cantilever b was ${\tilde{G}}_b=0.1$ MHz [43]. The frequency detunings of the two mechanical modes ${\Delta }_a$ and ${\Delta }_b$ were in the order of $GHz$, respectively. As shown in Figure 3a,b, the solid blue lines represent the evolution on the condition that $I/\left(g\chi \right)=0.5$, $\tau /\left(g\chi \right)=0.25$, while the solid red lines denote the evolution on the condition with parameters $I/\left(g\chi \right)=1,\tau /\left(g\chi \right)=0.5$. For the case in Figure 3c, the blue, green, and red lines represent the phase fluctuations with $\tau /\left(g\chi \right)=8\times {10}^{-3}$, $\tau /\left(g\chi \right)=0.001$, and $\tau /\left(g\chi \right)=0$, respectively. In Figure 3d, the red, blue and green lines denote the phase fluctuation with $\tau /\left(g\chi \right)=8\times {10}^{-2}$, $\tau /\left(g\chi \right)=0.01$, and $\tau /\left(g\chi \right)=0$. From the above analysis, we can demonstrate that amplitude fluctuations induce quadrature variance enhancement scaling as $g\chi$. However, our analysis also reveals that phase fluctuations at $\tau /\left(g\chi \right)=0.008$, $\tau /\left(g\chi \right)=0.001$ lead to significant enhancement of the $X_1$ quadrature variance, though a local minimum exists in the variance profile.

4. Discussion and Summary

Quantum squeezing represents a cornerstone of quantum optics and quantum information science, offering transformative potential for precision metrology, secure communication, and quantum computation. Despite its promise, significant challenges remain in generating and preserving squeezed states against decoherence and loss, as well as in developing efficient detection methods through homodyne measurement. This work bridges a critical gap in quantum information science by establishing a novel interface between atomic-scale quantum systems and macroscopic mechanical resonators. Our approach advances two key objectives: (1) the realization of mechanical squeezing for quantum information applications, where squeezed states enable precision measurements below the standard quantum limit (e.g., in gravitational wave detection) and enhance operational fidelity in quantum communication and computation; we present a theoretical framework for generating and controlling mechanical squeezing in hybrid systems, providing a scalable platform for non-classical state engineering; (2) the development of hybrid atom–mechanical systems for quantum information processing. The engineered coupling between Rydberg atoms and nanomechanical resonators creates a unique interface that combines the precision of atomic systems with the tunability of mechanical devices, offering promising avenues for quantum transduction and long-lived quantum memory implementations.

Also, we acknowledge that there are experimental challenges in our proposal. First, while the Rydberg atoms exhibit strong dipole moments and long coherence times (∼100 µs) in ultrahigh vacuum conditions, their sensitivity to electric field noise from charged cantilevers may necessitate advanced trapping techniques. Second, maintaining stable charges on nanoscale cantilevers is required, which may introduce technical hurdles, including charge fluctuations, which could destroy the atom–cantilever coupling. Furthermore, the thermal noise of the mechanical oscillation requires cryogenic cooling (<4 K) to make the system approach the quantum ground state. A particularly crucial point is that the coherence times of the mechanical cantilevers are about (∼ms), which are typically shorter than those of the Rydberg ensembles with the time scale (∼100 µs). To mitigate this mismatch, the implementation of hybrid pulsed-operation protocols combined with active feedback cooling of the cantilever system may be required.

In summary, we theoretically derived the nonlinear Hamiltonian and analytically studied the squeezing dynamics between two mechanical oscillators via coupling with the Rydberg atoms. This squeezing is induced by the classical drive on one of the single resonators. We also discussed the dynamics of squeezing in a dissipative system by solving master equations. For further applications, the atom and mechanical oscillator coupled system provides an ideal interface between the photons, which could be further generalized to quantum information processing. With the recent development in nano-manufacturing technologies, the realization of nanomechanical oscillators is becoming available, as this model provides a strong electric dipole–dipole coupling between the atom and the oscillators. Also, the long lifetime of the Rydberg atom provides us with enough time for coherent manipulation of the quantum state in the hybrid system. These characteristics position the hybrid system as a promising candidate for practical quantum information applications, particularly in the development of on-chip scalable quantum processors.