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Optomechanical cooling beyond the quantum backaction limit with frequency modulation

In the usual optomechanical cooling, even if the system has no thermal component, it still has a quantum limit-known as the quantum backaction limit (QBL)-on the minimum phonon number related to shot noise. By studying the side-band cooling regime in optomechanical system (OMS), we find that the cooling can be improved significantly when the frequency modulation (FM) that can suppress the Stokes heating processes is introduced into the system. We analyze and demonstrate the reasons of the phonon number below the QBL redefined in the whole stable region of the standard OMSs. The above analyses are further checked by numerically solving the differential equations of second order moments derived from the quantum master equation with broad system parameters, ranging from weak coupling (WC) to ultra-strong coupling (USC) and resolved side-band (RSB) to unresolved side-band (USB) regimes. Comparing with the cases of those without FM, the stable ground-state cooling can also be achieved even in the conventional unstable region.


I. INTRODUCTION
Over the past decades, special attention has been paid to studying the micromechanical resonators which are novel quantum devices and are used to explore various of quantum mechanics questions of the macroscopic scale, such as quantum-classical mechanics bound-ary [1][2][3], ultrahigh precision metrology [4,5], and gravitational-wave detection [6], etc.
However, for overcoming thermal noise to achieve those applications, the mechanical resonator needs to be cooled to its ground state at first. So far, various of proposals of cooling resonator have been proposed and discussed, including feedback control [7,8], modulating coupling strength [9,10], and cavity side-band cooling [11,12]. In these proposals mentioned above, cavity side-band cooling method is one of the most promising proposals, which has been studied widely and realized experimentally [13][14][15][16][17]. However, there has a QBL on the minimum phonon number related to shot noise even if the systems have no any coupling with thermal environment [12,18]. Recently, the QBL of cooling mechanical resonator has also been reached in cavity OMS via the side-band cooling approach [19]. It is natural to think that the cooling limit can be broken through some ways.
To this end, some theoretical proposals have been proposed to cool the mechanical resonator below the QBL, including dissipative coupling [20], pulsed cooling [21], electromagnetically-induced-transparency-like [22][23][24][25], trapping optical parametric amplifier [26], and injecting squeezed light [27], etc. And the proposal utilizing squeezed light to improve cooling has been proved experimentally [28]. Moreover, another interesting experiment also cools the mechanical resonator below the QBL via feedback-controllably engineering the pump field [29]. On the other hand, the modulated OMSs exhibit rich behaviors and have attracted amount of attention [30][31][32][33][34][35][36][37]. However, in those modulation proposals, the studying about mechanical resonator cooling is relatively rare. In Ref. [38], the authors study the optomechanical cooling theoretically using an adiabatic approximation when the frequency and damping of mechanical resonator are modulated periodically. As far as we know, the approach of directly using FM to improve mechanical cooling below the QBL has not yet been reported.
In this paper, we propose a proposal to improve micromechanical resonator cooling in OMS via modulating frequencies of both the optical and mechanical components. The FM of optical component is easy to implement and the modulation of micromechanical resonators has also been reported [38][39][40][41][42][43][44]. Here we provide a complete and simple understanding of the physical processes about improving mechanical cooling, which allows us to illustrate the deep reasons of the lower mechanical cooling. We consider a conventional OMS in which the mechanical resonator is coupled to optical cavity field via radiation pressure, and the cavity and mechanical resonator are modulated periodically. In such an OMS, we show that the Stokes heating processes can be fully suppressed via FM, and the physical mechanism is also explained and demonstrated through the Raman-scattering and frequency domain pictures. In addition, since the conventional QBL defined as in Ref. [45] is satisfied only in the WC region, we recalculate the QBL and steady-state final mean phonon number (MPN) of the standard OMSs in the absence of FM. Moreover, the dynamical evolution of MPN with FM is also obtained by numerically solving the differential equations of second order moments, which are derived strictly from the quantum master equation. And the results are shown respectively with very broad system parameters, e.g., the optomechanical coupling strength ranging from WC to USC and cavity decay rate ranging from RSB to USB regimes.
As we all know, the ground-state cooling cannot be achieved when either the coupling strength or cavity decay rate is too large in the standard OMSs without FM. However, the stable lower MPN can be achieved in the presence of FM even in the conventional unstable region of the standard OMSs. Different from the previous methods in Refs. [22][23][24][25][26][27], the mechanical resonator cooling is achieved below QBL only through the FM, which does not need extra technologies, such as trapping optical parametric amplifier, injecting squeezed light, or transparent window induced by auxiliary qubit, etc.
The paper is organized as follows: In Sec. II, we derive the linearized Hamiltonian of the OMS with synchronous FM and explain the physical mechanism of suppressing the Stokes processes through the Raman-scattering and frequency domain pictures. In Sec. III, firstly, we give the analytical expression of the steady-state final MPN in the absence of FM and discuss the QBL in the whole stable region. Then we study the dynamical evolution of MPN with broad system parameters through solving numerically the differential equations of second order moments derived from the quantum master equation. Lastly, a conclusion is given in Sec. IV.

II. SYSTEM AND HAMILTONIAN
We consider a generic OMS in which the mechanical resonator is coupled to a driven optical cavity via radiation pressure. The simplest possible cavity OMS model has been used to describe successfully most of the experiments to date. In the rotating frame at the driven laser frequency ω l , the Hamiltonian of the system is written as where a (b) and a † (b † ) represent the annihilation and creation operators for optical (mechanical) mode with frequency ω c (ω m ), respectively. ∆ c = ω c − ω l is the cavity laser detuning, g is the single photon optomechanical coupling strength [46], and E is the driving amplitude of laser. Following the usual linearization approach, a = a + δa = α + δa and b = b + δb = β + δb, the linearized Hamiltonian reads where ∆ c = ∆ c − g(β + β * ) and G = gα is linearized optomechanical coupling strength.
The Hamiltonian in Eq. (2) corresponds to the standard quantum Rabi model [47] including the beam-splitter interaction (Gδa † δb + G * δaδb † ) and two-mode squeezing interaction (Gδa † δb † +G * δaδb). In the usual side-band cooling mechanism, the resonant conditions of the beam-splitter interaction are an essential prerequisite to enhance (suppress) the anti-Stokes (Stokes) process, where the strict restrictions of small cavity decay and weak optomechanical coupling strength need to be satisfied. To break those restrictions, we introduce a cosine modulation of the free terms into the initial Hamiltonian. For simplicity, the modulation Hamiltonian is given by where ξ is the normalized modulation amplitude and ν is the modulation frequency. In the superconducting OMSs, the modulation of optical mode can be realized easily by tuning the magnetic flux in superconducting systems. Meanwhile, the modulation of mechanical mode can be realized by tuning the voltage of the gate electrode for graphene resonator [40,42,43], which can also be coupled superconducting qubit. In the presence of FM, we can derive the linearized Hamiltonian using the usual linearization approach (see Appendix A) To study the effect of FM on the system dynamics clearly, it is useful to perform the rotating transformation defined by where T denotes the time ordering operator. In the rotating frame defined by the transformation operator V 2 , the transformed Hamiltonian becomes Using the Jacobi-Anger expansions: e iξ sin(νt) = ∞ k=−∞ J k (ξ)e ikνt , the transformed Hamiltonian in Eq. (6) can be rewritten as where J k (ξ) is the Bessel function of the first kind with k being an integer. It is worth noting that, except for the usual linearized approach, our calculations have not taken any other approximations. The further understanding of the physics process can be explained in a Raman-scattering picture, as shown in Fig. 1, where |n, m denotes the state of n photons and m phonons in the displaced frame. Under the red detuning side-band resonant condition (∆ c = ω m ), the anti-Stokes process (the green arrows) is on resonance leading to enhance for the mechanical cooling. Different from the anti-Stokes process, the Stokes processes (the red arrows) exist the detuning (2ω m + kν) with coupling strength GJ k (ξ), which are able to be modulated independently by choosing appropriate parameters ξ and ν. Naturally, if the ratio GJ k (ξ)/(2ω m + kν) can be reduced significantly by modulating those two parameters, the heating (Stokes) process will be suppressed efficiently. However, in the absence of FM, the ratio is a constant G/(2ω m ), which implies that the efficient mechanical motion cooling needs to satisfy the restriction condition G 2ω m to suppress the Stokes heating process.
Furthermore, the physical mechanism of improving cooling can also be understood by using the frequency domain interpretation, as shown in Fig. 2, where the Stokes heating processes (the red peaks) are discrete and have been separated with modulation frequency ν space. For a given value of ν, there exists a corresponding k = k 0 to make sure the detuning (2ω m + k 0 ν) minimum in all side-bands of Stokes. If the modulation frequency ν is large enough, the other Stokes processes will be far away the resonant condition and the corresponding photons also cannot exist in the cavity, which lead to those contributions to heating mechanical resonator negligible. The primary purpose is to adopt an appropriate ξ to reduce the ratio GJ k 0 (ξ)/(2ω m +k 0 ν) for suppressing the nearest resonant heating process,   which is the major obstacle for cooling the mechanical resonator. That is easy to achieve because there are always a series of parameters ξ to satisfy J k 0 (ξ) = 0, as shown in Fig. 3.
Therefore, the nearest resonant heating process will also be negligible. Different from the usual side-band cooling proposals, which rely on the RSB regime κ < ω m , the method we proposed can suppress the nearest resonant heating process efficiently through manipulating its coupling strength GJ k 0 (ξ) even in USB regime, namely, even if the photons of resonating to Stokes processes can exist abundantly in the cavity, the photons cannot interact with the mechanical resonator.
Based on the above analyses, all the Stokes heating processes can be suppressed via FM approach, which strongly indicates that the QBL of mechanical cooling can be broken in the presence of FM. Next, we will verify that how it can be broken via calculating the MPN.

MODULATION
In this section, we study the dynamical evolution behavior of the phonon number under the linearized Hamiltonian in Eq. (4). It is not necessary to calculate the whole density matrix ρ to obtain the dynamical evolution of MPN. Here, we just need to solve a set of differential equations about the mean values of all the second order moments δa † δa , δb † δb , δa † δb , δaδb , δa 2 , and δb 2 , which can be derived via the quantum master equatioṅ where where ∆ = ∆ c + 1 2 ξν cos(νt) and Ω m = ω m + 1 2 ξν cos(νt) are the renormalized parameters. It is worth noting that the dynamical evolution of MPN is exact via solving the differential equations and the dynamics dimension is also not limited as the original quantum master equation. The system stability can also be determined by observing the dynamical behaviors of MPN in the presence of FM.

A. Cooling Limits in the absence of frequency modulation
Firstly, as a comparison, we give the quantum cooling limit for the whole stable region in the absence of FM (ξ = 0), which can be derived from Eq.
where the first term represents the classical cooling limit corresponding to the thermal environment, while the second term represents the quantum cooling limit, as the redefined QBL, corresponding to the cavity decay and quantum backaction. Under the red side-band resonant condition, the conventional steady-state final MPN is defined as (see Appendix B) with where A − (A + ) is the emitting (absorbing) rate of phonon and Γ opt is the cooling rate.
Different from n f , which requires very weak optomechanical coupling strength (see Fig. 4), the expression of Eq. (10) is satisfied in the whole stable region (|G| 2 < ω 2 m /4+κ 2 /16), which can be derived via Routh-Hurwitz criterion [49]. In order to clearly compare the two steadystate final MPNs with the dynamical results, we numerically solve the differential equations (Eq. (9)) in the absence of FM and show those results in Fig. 4. Here, for the dynamical results, we have assumed that the MPN of initial system equals to the average phonon number in thermal equilibrium bath δb † δb (t = 0) = n th = 10 3 , and the other second order moments are zero. The red solid line is the analytical steady-state final MPN which is given by Eq. (10), the black dashed line is the exactly numerical result which is obtained by solving Eq. (9) for the steady-state system, and the blue dot line is the conventional steady-state final MPN which is only satisfied with the very weak optomechanical coupling strength, as shown in Fig. 4. In addition, we also give the relationship between the steady-state final MPNs and cavity decay rate, as shown in Fig. 5. The results indicate that the ground-state cooling of mechanical resonator cannot be achieved when the cavity decay rate is too large or too small in the standard OMSs, which is consistent with previous studies.
In the whole stable region, we give a more general result of the cooling limit in the standard OMSs without FM, which can describe the mechanical cooling better in the broad parameter region. Next, we study the phonon number via solving the differential equations in the presence of FM, which will indicate that the cooling limit can be broken due to the existence of FM. For the WC regime, the ground-state cooling of mechanical resonator has been studied extensively in the RSB regime [50][51][52][53]. When the decay rate of the cavity is small enough (κ ω m ), the Stokes heating processes can be suppressed well in the usual red side-band cooling regime without FM. Therefore, the further suppression effect is unobvious via FM, namely, the improving rate of introducing the FM is very small. The result can be verified by showing the dynamical evolution of MPN in Fig. 6. One can see that there is not obvious difference between modulation and no modulation for WC and small cavity decay rate (see Table I). However, for larger coupling strength G = 0.1ω m and cavity decay κ = 0.1ω m , the MPN with FM is lower obviously than that without FM (see the cyan diamond line and magenta hexagram line in Fig. 6). This is due to the fact that the Stokes heating processes cannot be suppressed fully for larger coupling strength and cavity decay in the absence of FM.
In a word, the further cooling from FM is unobvious compared with the usual side-band cooling for the weaker optomechanical coupling in RSB regime. Based on the above analysis, we can easily infer that the cooling approach of utilizing FM to suppress Stokes scattering will stand out with the increment of optomechanical coupling strength and decay rate.

C. Strong coupling
In the SC regime (|G| > κ), the improving rate becomes obvious no matter the decay rate of cavity is large or small. This is very easy to understand based on the above analysis. It is worth noting that the standard OMSs without FM exist the dynamical stability condition (|G| 2 < ω 2 m /4 + κ 2 /16). Therefore, the SC region can be roughly divided to the stable (|G| < 0.5ω m ) and unstable (|G| > 0.5ω m ) regions, where the MPN of standard OMSs will converge or diverge corresponding to different regions in the absence of FM. Here, we should point out that the OMS with FM can be stable even in the unstable region mentioned above, which can be determined by observing the dynamical behaviors of MPN. Next, we study the mechanical resonator cooling in two different coupling regions, respectively.
In the stable region of the standard OMSs, we show the dynamical evolution of MPN corresponding to different optomechanical coupling strengths in Fig. 7  (see Table I). We can also see that the final MPN without FM increases with the increase of coupling strength. One reason is the saturation effect of the cooling rate [54] and the another is the enhanced Stokes processes. However, the final MPN with FM almost does not change with the increase of coupling strength. This is because the small decay rate of the optical cavity limits the final mechanical resonator cooling in the SC regime.
In the unstable region of the standard OMSs, where the ground-state cooling of mechanical resonator cannot be achieved due to the divergent behavior of phonon number (see G > 0.5ω m in Table I), we just show the dynamical evolution of MPN in the presence of FM corresponding to different optomechanical coupling strengths and modulation frequencies, as shown in Fig. 8. The results show that the system is stable due to the existence of FM even in the conventional unstable region. With the increase of modulation frequency, the system becomes more stable and the final MPN is smaller. The fundamental reason is that the large modulation frequency can suppress the Stokes heating processes efficiently. Furthermore, we also note that, for the small modulation frequency in Fig. 8, the final MPN increases with the increase of optomehcanical coupling strength. This is because the small modulation frequency cannot suppress the Stokes heating processes fully when the coupling strength is very large. Moreover, we also show the dynamical evolution of MPN in the USC regime (|G| > ω m ), as shown in Fig. 8. We find that the ground-state cooling can also be achieved in USC regime with FM. So we can infer that, for more larger coupling strength, more larger modulation frequency will be needed to achieve the better ground-state cooling of the mechanical resonator.

D. Unresolved side-band regime
For the USB regime (κ > ω m ), the Stokes processes cannot be suppressed well in the standard OMSs, which results in an unsatisfactory ground-state cooling of mechanical resonator (see Table I). However, in the presence of FM, the nearest resonant Stokes process can be fully suppressed and the other Stokes processes can also be neglected when the modulation frequency is large enough. Similarly, we show the dynamical evolution of MPN corresponding to different cavity decay rates in Fig. 9. We find that, in the absence of FM, the ground-state cooling cannot be achieved when the cavity decay rate is too large.
However, in the presence of FM, the ground-state cooling can be achieved very well even in the USB regime.
In the above discussions, the study of the MPN dynamical evolution is only related to one of the system parameters, i.e., the optomechanical coupling strength or cavity decay rate.
To clearly demonstrate the results of their synergistic interaction, we plot the average value of the final MPN after the system is stable, as shown in Fig. 10, where the final stable MPN changes with the optomechanical coupling strength and cavity decay rate, with or without the FM. In the presence of FM, the final MPN of the mechanical resonator is lower than that without FM in the stable region. In addition, we can also achieve the ground-state cooling of mechanical resonator even in the conventional unstable region, as shown the left side of the yellow plane in Fig. 10. The results indicate that the system with FM is still stable and the ground-state cooling can also be achieved even in the conventional unstable region for the standard OMSs.

E. Small modulation frequency
In the above subsections, in order to suppress the Stokes heating processes better, the large modulation frequency is necessary. In practice, however, too large modulation frequency may not be conducive to experimental implementation, although various of schemes about the FM of mechanical resonator have been reported theoretically and experimentally [38][39][40][41][42][43][44]. Therefore, we consider the effect of a small modulation frequency on the system in this subsection. We set the modulation frequency as ν = 2ω m , where one of the Stokes processes (k 0 = −1) can be resonant ideally like the anti-Stokes process in the red detuning side-band resonant regime. Here, the resonant Stokes process can also be sup-  processes (k = −1) cannot be suppressed well due to J k =−1 (3.8317) = 0 at this time. The dynamical evolution of MPN with the small modulation frequency is shown in Fig. 11 for different system parameters. The results indicate that the cooling effect resorting to FM is still better than the standard OMSs without FM under these system parameters. The influences of MPN by changing system parameters are also similar to the situation of large modulation frequency.

F. Asynchronous frequency modulation
In previous discussions, we have assumed that the modulations of optical and mechanical components are identical and synchronous to demonstrate clearly the physical mechanism of the further cooling via FM. Here, we will study the effects of asynchronous FM, e.g., different amplitudes, phases, and frequencies of the modulation. The Hamiltonian of the modulation parts changes to H M = [ξ 1 ν 1 cos(ν 1 t)a † a + ξ 2 ν 2 cos( and ν 1 (ν 2 ) are normalized amplitude and frequency of the optical (mechanical) modulation, respectively, and θ is the relative phase of two modulations.
Firstly, we study the effects of different optical and mechanical modulation amplitudes, where the other modulation parameters are identical, i.e., ξ 1 = ξ 2 , θ = 0, and ν 1 = ν 2 . The , and the other parameters are the same as in Fig. 6.
interaction Hamiltonian can be written as Based on the previous analyses, the mechanical cooling can be improved if the nearest resonant Stokes heating process is suppressed with (ξ 1 + ξ 2 )/2 = ξ 0 , where J k 0 (ξ 0 ) = 0. It is worth noting that the beam-splitter interactions are also related to the different modulation amplitudes. Therefore, the improving mechanical cooling cannot be achieved if the difference value (ξ 1 − ξ 2 )/2 is close to ξ 0 , which greatly reduces the optomechanical coupling. In addition, the similar results can also be obtained with θ = π, where the beam-splitter and two-mode squeezing interactions exchange their coupling strengths. The dynamical results of MPN are shown in Fig. 12 with different amplitudes.
For the different phases of FM, we also study the effects by changing θ and show the results in Fig. 13. We find that the improving mechanical cooling can also be achieved when θ is far from π with ξ 1 = ξ 2 and ν 1 = ν 2 . That is easy to understand due to the exchange of coupling strengths for Eq. (13) when the relative phase θ equals to π, where the coupling strength of the nearest resonant two-mode squeezing interaction is GJ 0 (0) = G.
And the Stokes processes cannot be suppressed so that the improving mechanical cooling is not achieved. Lastly, we study the effects of different modulation frequencies, i.e., ν 1 = ν 2 . We find that the improving mechanical cooling cannot be achieved when the two modulation frequencies satisfy |ν 1 − ν 2 | 2ω m , as shown in Fig. 14 where a (b) and a † (b † ) represent the annihilation and creation operators for optical (mechanical) mode with the corresponding frequency ω c (ω m ), respectively. The parameter g is the single photon optomechanical coupling rate. The parameters E and ω l are the driving amplitude and frequency, respectively. By performing a rotating transformation defined by where ∆ c = ω c − ω l is the cavity laser detuning.
The quantum Langevin equations are given bẏ where ∆ c = ∆ c − g(β * + β) is the effective detuning modified by optomechanical coupling, G = gα is linearized optomechanical coupling strength, and we have neglected the nonlinear terms igδa(δb † +δb) and igδa † δa due the strong coherent driving conditions. Then we obtain the linearized Hamiltonian (see Eq. (4) in the main text) in the presence of FM.
where we can obtain the optical force F = (Gδa † +G * δa)/x ZPF . The quantum noise spectrum of the optical force is given by the Fourier transform of the autocorrelation function S F F (ω) = dte iωt F (t)F (0) , which is calculated easily in the frequency domain. In the WC regime, the optomechanical coupling can be regarded as a perturbation to the optical field, where the spectral density of the optical force can be calculated properly without considering optomechanical coupling. In the frequency domain, the cavity mode fluctuation operator δa(ω) can be obtained in the absence of the optomechanical coupling, (B2) According to the Hamiltonian in Eq. (B1), we can write the rate equations of the mechanical resonator aṡ P n = Γ n+1→n P n+1 + Γ n−1→n P n−1 − Γ n→n−1 P n − Γ n→n+1 P n +γ(n th + 1)(n + 1)P n+1 + γn th nP n−1 − γ(n th + 1)nP n − γn th (n + 1)P n , (B3) where P n is the occupation probability of the mechanical Fock staet |n with n phonons.
Γ n→m is the transition rate of phononic state from |n to |m induced by the optomechanical coupling. Using the Fermi golden rule and the above calculations, we can obtain where A − (A + ) is the emitting (absorbing) rate of phonon. By solving Eq. (B3) withṖ n = 0, the steady-state final MPN can be obtained as In the absence of intrinsic mechanical damping rate (γ = 0), we also obtain the fundamental quantum limit of cooling