Improving mechanical squeezing in a dissipative optomechanical system with quadratic dispersive coupling

We theoretically investigate that mechanical quantum squeezing in a dissipative optomechanical system, driven by two driving tones, can be improved through quadratic dispersive coupling (QDC) which appears by adjusting a membrane in an appropriate position of an optical cavity. The analytical expressions are derived for revealing the intrinsic mechanism of the mechanical squeezing improvement. Remarkably, compared with the case without QDC, an optimal condition involved the QDC is found to considerably enhance the mechanical squeezing, without reducing the purity of the mechanical squeezed state which even increases at low temperature. We also show that, in this scheme, the mechanical squeezing is still improved by QDC and beyond 3 dB even though the temperature rises. This improvement effect can be reflected by the broader frequency band of the measurable squeezing output field from the cavity. This provides a new opportunity to explore high-precision measurements and quantum nature of macroscopic objects.


Introduction
The generation of quadrature-squeezed state of macroscopic mechanical oscillator is a promising technique to realize ultrasensitive force measurements [1], continuous variable quantum-information processing [2], and the exploration of quantum-classical boundary [3]. This kind of squeezing, i.e. reducing the uncertainty of one motional quadrature below the zero-point level, can be realized by exploiting a spring constant tuned silicon microcantilever [4]. With significant progress of quantum technology, other strategies based on various systems have been studied, e.g. NV-center system [5] and trapped-ion system [6]. After all, this task may not always be implementable with current technologies, and therefore one has to consider a family of mechanical squeezing schemes realized in recent experimental setups.
Except for the dispersive type, the OMSs with dissipative coupling, in which the mechanical motion modulates the decay rate of the cavity, can be implemented by, e.g. a waveguide coupled to a microdisk resonator [32], a membrane-in-the-middle structure (MIMS) [33,34], a tapered fiber coupled to a whispering-gallery mode [35,36], and a photonic crystal cavity [37]. This alternative, less-explored coupling leads to the measurement sensitivity [38], the self-oscillation [39], the strong coupling effect [40], the optical spring effect [41], the squeezing of the output light [42], and the squeezing of the mechanical oscillator [43][44][45][46]. In particular, with the aid of two imbalanced driving tones, the dissipative OMS makes the mechanical squeezing achieve 3 dB [45]. Besides, quadratic dispersive coupling (QDC) also appears in OMS, especially in a MIMS where the membrane is placed in an appropriate position [47][48][49][50]. Actually, it has been recently found that the QDC is accompanied by a dissipative coupling [33,51].
Inspired by the idea in [45], we propose a feasible scheme for improving mechanical squeezing with the aid of QDC in a dissipative OMS injected by two-tone driving. Our results manifest that this strategy, simultaneously exploiting the dissipative and QDCs, produces stronger squeezing in displacement fluctuation of membrane due to a cooperation-based enhancement from two-tone driving and QDC. Based on our analytical result, the displacement squeezing exceeds the one solely by using two-tone driving since the QDC further reduce the unwanted fluctuation amplification from blue-detuned driving. The enhancement conditions are also displayed. In addition, the broader frequency band, in comparison with the case without QDC, to measure output squeezing spectrum from optical cavity indicates the improvement of the squeezing effect.
The rest part is structured as follows. In section 2, the theoretical frame is proposed. In section 3, we calculate the steady-state variances of displacement operator for analyzing the quantum squeezing. In section 4, we first find the analytical result to reveal the intrinsic mechanism of the squeezing enhancement from QDC, and show the enhancement by comparing with the case in the absence of QDC. In section 5, we show the detection of the squeezing improvement via the output field. Finally, this paper is summarized in section 7.

Theoretical framework
To implement this strategy, we consider an OMS based on MIMS where a thin membrane is placed in an optical cavity. The membrane is then treated as a mechanical oscillator with effective mass m, resonance frequency ω m and damping rate γ, and its mechanical oscillation can be described by displacement and momentum operators, q and p. On the one hand, the mechanical oscillation has an influence on the cavity resonance frequency ω c (q), resulting in dispersive optomechanical coupling. The QDC even dominates if the membrane is placed at the center of the node or antinode of the intracavity mode [48,52]. On the other hand, the membrane position also modulates the cavity damping rate κ(q), leading to dissipative optomechanical coupling [53]. Furthermore, a two-tone laser, including blue-and red-detuned fields with different frequencies ω ± and powers P ± , is considered to drive the cavity mode. The Hamiltonian of system then reads where c is the annihilation operator of intracavity mode and ε ± = P ± /ℏω ± is the driving field amplitude. c in denotes the input vacuum noise operator, satisfying with zero mean ⟨c in ⟩ = 0, where ⟨•⟩ stands for the ensemble average. For small displacement, ω c (q) and κ(q) are expanded as and g ′ κ denote the linear dispersive coupling, QDC and dissipative coupling strengths, respectively. By reexpressing the displacement and momentum of mechanical oscillator with the associated coupling strengths g ω1 = q zp g ′ ω1 , g ω2 = q 2 zp g ′ ω2 , and g κ = q zp g ′ κ , and the zero-point fluctuation of the mechanical oscillator q zp = ℏ mωm . Applying this Hamiltonian into the quantum Langevin equation, we have the motion equations of both modeṡ where thermal noise operator b in of mechanical oscillator satisfies the correlators as in the limit ω m /γ ≫ 1 [54]. n th = exp ℏωm is the thermal phonon number of mechanical oscillator, with the Boltzmann constant k B and environment temperature T.
Due to the two-tone laser fed into the cavity, we can split both modes into a steady-state amplitude and a fluctuating term as c = c s+ exp(−iω + t) + c s− exp(−iω − t) + δc and b = b s + δb, where c s± and b s are steady-state amplitudes for optical and mechanical modes and δc and δb are their corresponding fluctuation operators. The driving frequencies are set as ω ± ≈ ω c + Q s g ω1 Based on dynamical equation (4), we obtain the steady-state amplitudes as with the mixed dispersive coupling strength g Ω = g ω1 + Q s g ω2 . Obviously, the steady amplitudes nonlinearly reply on each other. However, if ω m is large relative to other parameters, one can approximately obtain the decoupled steady amplitudes as On the other hand, we only keep the first-order fluctuating terms in dynamical equation (4) and leave their linearized equations of motion, where we have ignored the counterrotating terms. It is noted that the terms of δb † and δc † , in equations (9) and (10) respectively, only result from the blue-detuned tone and have been shown to make contribution to mechanical squeezing [45]. Moreover, due to the cooperation between QDC and two-tone laser, a term of δb † in equation (10) will be demonstrated to further enhance mechanical squeezing in the following sections.

Quantum fluctuations
Let us introduce the quadrature components of both optical and mechanical fluctuations as and the associated quadrature components of the input noise operators as Based on equations (9) and (10), these observables are governed bẏ with the operator vector Θ(t) = (δX, δY, δQ, δP) T , the drift matrix and the vector of all noise operators where the coefficients in equation (14) are given by The system is stable if the stability condition is fulfilled, as shown in appendix A. Based on equation (13) and Fourier transform , the fluctuations of the displacement and momentum of the mechanical oscillator can be solved in the frequency domain, Based on the spectrum for any fluctuation operator δO with correlators of the input noises (2) and (5) in the frequency space, the spectra of both fluctuations of mechanical oscillator can be obtained as Unfortunately, the exact forms of equations (17) and (18) seem too fussy to reveal the intrinsic mechanism of mechanical squeezing enhancement from QDC. Thus, according to the relevant conditions, we first find the approximate, but analytical, solution for displacement fluctuation δQ(ω) to exhibit the intrinsic mechanism, which will be found to meet the strict numerical results in the following. Based on the detailed derivation in appendices B and C, the approximate coefficients A j in equation (17) are given by . The steady-state variances ⟨δQ 2 ⟩ and ⟨δP 2 ⟩ in the displacement and momentum operators of the mechanical oscillator are defined by In the following, we focus on ⟨δQ 2 ⟩ for investigating the squeezing of mechanical displacement. Without any optomechanical couplings, the result for mechanical oscillator in thermal state is ⟨δQ 2 ⟩ = n th + 1/2, and thus turns to ⟨δQ 2 ⟩ g = 1/2 at zero temperature, i.e. in ground state. In the whole manuscript, we define transformation −10 log 10 ⟨δQ 2 ⟩/⟨δQ 2 ⟩ g in the dB unit to indicate the mechanical displacement squeezing degree. The squeezing condition ⟨δQ 2 ⟩ < ⟨δQ 2 ⟩ g is then converted to −10 log 10 ⟨δQ 2 ⟩/⟨δQ 2 ⟩ g > 0.
Applying equation (20), we obtain the variance of Q as where each term V j corresponds the term relevant to coefficient A j in equation (20), with j = X, Y, Q, P. Thus, V X and V Y are resulted from the amplitude quadrature X in and phase quadrature Y in of the input electromagnetic vacuum noise, while the terms V Q and V P are associated with the amplitude quadrature Q in and phase quadrature P in of the input mechanical thermal noise. Using the analytical forms of corresponding  (24) and (25) coefficients (22) with γ ≪ κ, we find that the dominant terms of coefficients V j are approximately expressed with the analytical forms with the coefficients In equation (25), while ζ ∆ = (ε 2 − − ε 2 + )g 2 κ and ε ± denote the effect from two imbalanced driving tones, indicates the effect from QDC. Thus, equation (25), as our important result, reveal the intrinsic mechanism of improving mechanical squeezing by exploiting QDC, and accord with the exact results shown in figure 1.
Besides, we also provide the purity of the squeezed mechanical state defined as 1 1+2n eff [55], where the effective thermal occupancy of the mechanical oscillator n eff is given by Here the ⟨δQ 2 ⟩ and ⟨δP 2 ⟩ are defined by equation (23) and the correlation ⟨{δQ, δP}⟩ is given by with Therefore, n eff , as well as the purity, completely depends on δQ(ω) and δP(ω) in equations (17) and (18) which we have obtained via the coefficients (C4) and (C6) (in appendix C). According to this definition, the value of the purity is between zero and unity, and reaches the maximum (i.e. unity) if the mechanical oscillator is in a pure squeezed vacuum state. Obviously, the value of the purity decreases as n eff increases, indicating that the mixedness of the mechanical state increases. Thus, we will analyze the purity of the squeezed mechanical state via n eff in the following.

Improvement to mechanical squeezing
On the one hand, the two-tone driving generates the mechanical squeezing in spite of the detrimental effect from blue-detuned tone. Due to the coefficient ζ ∆ = (ε 2 − − ε 2 + )g 2 κ in V Q and V P (equations (25c) and (25d)), blue-detuned tone heats the mechanical motion via weakening the cooling effect of red-detuned tone, and the factor (ε − + ε + ) 2 in V Y (equation (25b)) also manifests that blue-detuned tone increases the fluctuation contribution from the phase quadrature of vacuum noise Y in . However, it is still noted that the factor (ε − − ε + ) 2 in V X (equation (25a)) oppositely makes blue-detuned tone beneficial to reduce the contribution from the amplitude quadrature of vacuum noise X in . Thus, the mechanical squeezing still achieves, i.e. ⟨δQ 2 ⟩ < 1/2 in equation (24), as long as the reduction effect to V X is significantly stronger. It means that, for achieving optimal squeezing, two powers of two driving tones should be appropriately tuned.
On the other hand, we demonstrate a scheme to improve the mechanical squeezing via reducing the fluctuations from Y in and P in that, based on equations (25b) and (25d), rise just due to the introduction of the two-tone driving. In our proposal, the QDC, indicated by Υ + = ε 2 − + ε − ε + + ε 2 + g ω2 /ω 2 m in equations (25b) and (25d), is considered as a key to suppress the detrimental effect from the two-tone driving. In the following, we discuss the improvement effect from QDC in low and high temperature regimes.

At low temperature
The fluctuation from P in , as well as Q in , can be omitted at low temperature. It means that the fluctuation from Y in in equation (25b) becomes the major contribution to suppress the mechanical squeezing and should be minimized via QDC. According to equation (25b), if Υ + fulfills that is, the QDC strength meets the condition as the vacuum fluctuation contribution from Y in can be optimized as and thus is effectively suppressed, as well as the total variance ⟨δQ 2 ⟩ in equation (24). In this scenario, because of the introduction of QDC and the optimization via the relevant condition equation (31), one is capable of improving the squeezing of the mechanical displacement, in comparison with the scheme in the absence of QDC. For numerical analysis, we choose the parameters inspired by [32,45]. The frequencies of the red-and blue-detuned tones are ω − = 2πc/λ − and ω + = ω − + 2ω m , respectively, with light wavelength λ − = 1564.25 nm and the speed of light c. The effective mass of mechanical oscillator is m = 2 pg, and the frequency ω m = 2π × 25.45 MHz. The cavity decay rate is κ = 0.05ω m while mechanical quality factor is Q = ω m /γ = 5000. With these parameters, we report the numerical results of mechanical squeezing via the squeezing level −10 log 10 ⟨δQ 2 ⟩/⟨δQ 2 ⟩ g in the dB unit.
In figure 1, we study the squeezing level of mechanical quadrature ⟨δQ 2 ⟩ as a function of the laser power ratio P + /P − , with P − = 18 µW, g κ = −2π × 26.6 MHz nm −1 × q zp = −3.05 kHz, and g ω2 = −1 Hz, at zero temperature (more precisely, at low temperature where the mechanical mode is near to ground state). The results from analytical forms (25) (marked by circle and triangle) are excellently consistent with the exact results (solid lines). As shown, while the specific blue-detuned laser, combined with the red-detuned laser, is exploited to generate mechanical squeezing (solid thin black line) [45], a further enhancement of the squeezing is still evident in a wide power range due to the present of QDC (solid thick orange line). The highest squeezing level is about 6.3 dB in the absence of QDC at P + /P − ≈ 0.63, but boosting up to about 10.6 dB in the presence of QDC at P + /P − ≈ 0.85. Just as the analysis of equation (25) above, the fluctuation from Y in would have increased due to the presence of two-driving power, but g ω2 can suppress this fluctuation via optimal condition equation (31) and enhances the mechanical squeezing eventually. In addition, two dashed lines reflect that small linear dispersive coupling does not dramatically change the results, and thus we omit this aspect in the following.
In order to compare the cases without QDC (g ω2 = 0) and with the optimal QDC based on equation (31), we further present their squeezing levels as a function of two laser powers P ± in figure 2. As shown in figure 2(a), the maximum squeezing reaches about 6.3 dB in the absence of QDC and, due to the cooperation of two driving tones, a region beyond 6 dB can be established. On the other hand, as depicted in figure 2(b), the optimal QDC generates squeezing up to 10 dB in a specific region where the maximum even reaches about 10.7 dB. Furthermore, the dashed line in figure 2(b) shows that, due to the optimal QDC, the region in which the squeezing surpasses the maximum point of the case without QDC is significantly extended. In other words, QDC brings stronger mechanical squeezing when meeting the optimal condition with the driving powers. Figure 3 further report how g ω2 improves the mechanical squeezing. In figure 3(a), we plot the optimal g ω2 as a function of two driving powers P ± based on equation (31). It manifests that, to achieve the maximum of mechanical squeezing, g ω2 should be chosen as g ω2 ≈ −0.93 Hz. The region II shows that, with specific range of P ± , there is a range of g ω2 around −1 Hz to reach high squeezing level as 10 dB. The region I shows a wider range of g ω2 that generates higher mechanical squeezing than the maximum in absence of QDC. Then in figure 3(b), we plot squeezing level as a function of ratio P + /P − and g ω2 . It further indicates that g ω2 around −1 Hz makes the squeezing as high as 10 dB via coordinating with P ± . In a range −5 Hz ≲ g ω2 < 0 and with appropriate P ± , the squeezing can obviously surpass the maximum without QDC. To sum up, the adjustment to g ω2 , e.g. by manipulating the membrane in an appropriate position, is useful to improve mechanical squeezing with the optimal powers P ± .
The displacement squeezing level and the effective thermal occupancy n eff of the mechanical squeezed state (equation (27)) are exhibited in figure 4 for various temperatures of the environment T = 0, 20, 40, and 60 mK when P + /P − ≈ 0.63. As seen in figure 4(a), the squeezing is lower with rising temperature, but with QDC (thick lines), the maximums are up to about 7.5, 5.4, 4.1, and 3.1 dB at T = 0, 20, 40 and 60 mK, respectively, stronger than the ones without QDC (thin lines). Besides, at these temperatures, the optimal red-detuned tone powers become P − ≈ 45, 120, 160, and 180 µW, respectively, implying a gradually rising trend. Especially, at T = 60 mK, the maximum still achieves 3 dB in the presence of QDC (thick dot-dashed line), but it is impossible without QDC (thin dot-dashed line). On the other hand, when P − is set to reach the maximum squeezing, as shown in figure 4(b), in the presence of QDC effective thermal occupancy n eff achieves a minimum close to 0.04 at T = 0 K, distinctly lower than 0.07 in the absence of QDC, which means the rise of the purity 1 1+2n eff . This lower-n eff situation in the presence of QDC is still obvious at T = 20 mK, but with increasing temperature, the difference between both cases with and without QDC becomes subtle. Notably, n eff rises more rapidly with increasing P − in the presence of QDC. This reflects that QDC does not damage the purity of the squeezed mechanical state with suitable driving powers, but even reinforces the purity at low temperature. Figure 5 shows how the squeezing level of mechanical displacement varies with temperature for different red-detuned tone powers P − = 50, 100, 200, 300 µW. Despite the damage from temperature, the presence of QDC still enhances the squeezing at the same temperature. Besides, in the case with QDC, temperature also alters the optimal driving powers since the squeezing for higher P − decreases less rapidly with temperature. At T = 0 K, the squeezing level is about 7.6 dB for P − = 50 µW, close to the one for P − = 100 µW, but observably higher than 6.7 dB for P − = 200 µW and 5.6 dB for P − = 300 µW. While at several mK the squeezing for P − = 100 µW immediately surpasses the one for P − = 50 µW, at T ≳ 30 mK the squeezing for P − = 200 µW begins to exceed the one for P − = 100 µW. The squeezing for P − = 300 µW is too low to surpass the one for P − = 200 µW. As shown, the enhancement effect from QDC becomes more feeble at higher temperature with the optimal P − . In other words, lower temperature not only generates stronger mechanical squeezing, but also makes the enhancement effect of QDC more apparent.

At high temperature
The fluctuations from input mechanical thermal noise, represented by equations (25c) and (25d), become important at high temperature of environment. Thus, both contributions from Y in (equation (25b)) and P in (equation (25d)) should be considered together. Their total contribution is represented by where the coefficients E ′ j with j = 0, 1, 2 are given by relying on the temperature of environment via thermal phonon number of the mechanical oscillator n th . Even though the coefficients are different and the temperature of environment obviously reduces the mechanical squeezing, the expression equation (33) reveals that this part of the fluctuation can still be optimized through QDC, Υ + = ε 2 − + ε − ε + + ε 2 + g ω2 /ω 2 m , based on the same way in section 4.1. The optimal condition is given by similar with the condition equation (30), but influenced by temperature. Therefore, the improvement effect from QDC still works in high temperature regime.  The figure 6 depicts the degree of the displacement squeezing (dB) versus driving power ratio P + /P − for n eff = 10 3 , 5 × 10 3 , and 10 4 , corresponding to temperature T = 1.2 K, 6.1 K, and 12.2 K. The mechanical squeezing is suppressed when temperature becomes higher. However, the QDC is found to improve the squeezing effect with appropriate driving powers at each temperature by comparing the corresponding thin and thick lines in figure 6. It is noted that, at high temperature, the mechanical squeezing is lower than 3 dB without QDC (thin dotted line), but can surpass 3 dB due to the optimization from QDC (thick dotted line).

The detection of mechanical squeezing
Based on the input-output boundary condition c out = 2κ(q)c − c in , the fluctuation of the output light field δc out is introduced as to detect the squeezing of the mechanical oscillator. If adopting homodyne detection, we then define the output operator  with the quadrature operators X out = (δc out + δc † out )/ √ 2 and Y out = −i(δc out − δc † out )/ √ 2, and a controllable phase ϕ. By solving equation (13) in frequency domain, the output operator z out is given by with the coefficients J j (j = X, Y, Q, P). As a consequence, by combining equation (19) with equations (2) and (5), the spectrum of the output field z out is written as Instead, we will use −10 log 10 [S out (ϕ, ω)/S out (ϕ, ω) vac ], in the unit of dB, as a squeezing degree of measured light field spectrum S out (ϕ, ω) to reflect mechanical squeezing. Here S out (ϕ, ω) vac = 1/2 for vacuum state. In figure 7 we study S out (ϕ, ω) (dB) as functions of normalized frequency ω/ω m when T = 0. As expected, as long as laser powers are set to optimize mechanical squeezing, shown in figure 2, the squeezing of the output fluctuation occurs, revealing the displacement squeezing of the mechanical oscillator. Comparing the cases with and without QDC (solid and dashed lines), we see that their maximum squeezing levels are close, up to 6 dB and 5.4 dB at ω = 0 respectively. More importantly for the case with QDC, the squeezing of the output is detectable in a broader frequency band, up to |ω| ≲ 0.1ω m , in comparison with |ω| ≲ 0.024ω m for the case without QDC. It is the broader band of measuring output squeezing that hints the improvement effect of QDC to mechanical squeezing. In figure 8 the output spectrum S out (ϕ, ω) (dB) is plotted as a function of normalized frequency ω/ω m and phase ϕ when T = 0. Under the optimal condition of two driving powers, the optimal phases for squeezing maximum are different between the cases without and with QDC, chosen for depicting figure 7. It is noted again that the broader frequency band of displaying output squeezing indicates the mechanical squeezing reinforcement from QDC.

Possible experimental implementation
Strong dissipative coupling has been reported in several setups recently [32-34, 51, 53, 56-58]. One of the schemes to obtain dissipative coupling and QDC simultaneously is using a Fabry-Perot cavity with a membrane inside [51]. As shown in figure 9, a Fabry-Perot cavity is split into two sub-cavities by a membrane with transmission |t m |. The two sub-cavities, with lengths L 1 and L 2 and decay rates κ 1 and κ 2 , are degenerate at frequency ω 0 = N 1 πc/L 1 = N 2 πc/L 2 for integers N 1 and N 2 and coupled to each other via the membrane at rate J = c|t m |/2 √ L 1 L 2 . On the other hand, the membrane is treated as an oscillator with mechanical displacement q. According to [51], this membrane-cavity system has two eigenmodes with eigenfrequencies and corresponding decay rates with sub-cavity optomechanical coupling G j = (−1) j ω 0 /L j ,κ = (κ 1 + κ 2 )/2, and ∆κ = (κ 1 − κ 2 )/2. Thus, the dependence on membrane displacement q leads to the linear dispersive coupling which vanishes at membrane position with G 1 < 0 and G 2 > 0. Importantly, the QDC is given by which becomes if q meets equation (43). Here G j = (−1) j ω 0 /L j and J = c|t m |/2 √ L 1 L 2 with the transmission coefficient of membrane t m . In the similar way, equation (41) leads to the dissipative coupling which becomes if q meets equation (43). The detailed derivation in this section has been shown in [51]. Interestingly, equations (45) and (47) manifest the coexistence of QDC in a dissipative QMS without introducing linear dispersive coupling. As for our scheme, we focus on the coupling rates as Figure 9. Schematic of an optomechanical setup based on membrane-cavity system. A Fabry-Perot cavity is split into two sub-cavities, with lengths L1 and L2 and decay rates κ1 and κ2, by a membrane with transmission |tm|. The two sub-cavities are coupled to each other via the membrane at rate J = c|tm|/2 √ L1L2. The membrane is treated as an oscillator with mechanical displacement q.

Conclusion
In summary, we have found that QDC can substantially improve squeezing effect on the mechanical oscillator in a dissipative QMS driven by two-tone laser. We have provided an analytical form of steady-state displacement variance to illustrate the intrinsic mechanism of the improvement. This mechanism indicates that, in displacement fluctuation, while blue-detuned driving tone is exploited to generate mechanical squeezing because its reducing effect on the fluctuation from amplitude quadrature of optical input noise exceeds its rising effect on the fluctuation from the corresponding phase quadrature, the QDC is capable of reducing the contribution from the corresponding phase quadrature instead and further enhancing the squeezing effect. We exhibit that the mechanical squeezing is actually boosted by an optimal condition of QDC in cooperation with appropriate driving powers. As shown, even at finite temperature, QDC can also improve the squeezing without losing the purity of the mechanical squeezed state, but even rising the purity at low temperature. In addition, this advantage from QDC can be detected since the squeezing of the output field from the cavity is measurable in a broader frequency band, in comparison with the case without QDC. The squeezing enhancement of mechanical motion may reveal a new possibility for implementing ultrasensitive displacement measurement and exploring quantum nature of macroscopic object.

Appendix B. The approximate drift matrix and noise vector
Here we make some approximations to the proposed system without linear dispersive coupling (g ω1 = 0 or g Ω = Q s g ω2 ) to reveal the intrinsic mechanism of the mechanical squeezing enhancement stemming from QDC. Besides, we assume the system in good cavity regime (γ ≪ κ ≪ ω m ) and with larger red-detuned laser (ε − > ε + ). First, the approximations to the drift matrix (14), as well as the corresponding conditions, are listed below: (a) The elements M 11 and M 22 are approximately simplified as when the condition holds, according to Q s in equation (8). (b) According to equation (8) for c s± , we know Combined these equations with two conditions (ε − − ε + )/(ε − + ε + ) ≫ (κ/ω m )|Q s g ω2 /g κ | and κ/ω m ∼ |Q s g ω2 /g κ | ≪ 1, we further acquire the conditions as Im(c s− ± c s+ )|g κ | ≫ Re(c s− ± c s+ )|Q s g ω2 |.
Under these conditions, all first terms in equations (16a)-(16d), i.e. parameters U j± and V j± in matrix M, are their leading terms, and the rest terms can be neglected. These parameters are