Suppression of Stokes heating processes and improved optomechanical cooling with frequency modulation

Yang Bao,Qinghong Liao,2,Qingmin Zhao and Jing Wu

1 Department of Electronic Information Engineering,Nanchang University,Nanchang 330031,China

2 State Key Laboratory of Low-Dimensional Quantum Physics,Department of Physics,Frontier Science Center for Quantum Information,Tsinghua University,Beijing 100084,China

Abstract Ground-state cooling of mesoscopic mechanical objects is still a major challenge in the unresolved-sideband regime.We present a frequency modulation(FM)scheme to achieve cooling of the mechanical resonator to its ground-state in a double-cavity optomechanical system containing a mechanical resonator.The mean phonon number is determined by numerically solving a set of differential equations derived from the quantum master equations.Due to efficient suppression of Stokes heating processes in the presence of FM,the ground-state cooling,indicated by numerical calculations,is significantly achievable,regardless of whether in the resolved-sideband regime or the unresolved-sideband regime.Furthermore,by choosing parameters reasonably,the improvement of the quantum cooling limit is found to be capable of being positively correlated with the modulation frequency.This method provides new insight into quantum manipulation and creates more possibilities for applications of quantum devices.

Keywords:ground-state cooling,Stokes heating processes,frequency modulation,unresolved sideband

1.Introduction

A cavity optomechanical system[1–4]based on the radiation pressure generated by the interaction between the optical field and the mechanical mode provides a powerful platform for the exploration of quantum behavior in fundamental quantum physics[5–9].With regard to the results of the maturity of modern manufacturing technology and the proposals of various optical experimental equipment[10–14],the cavity optomechanical system has been developed like never before and has received tremendous interest in terms of both theoretical studies and experimental research.To date,the size of the mechanical resonators studied in the cavity optomechanical system has spanned from the microscopic to the macroscopic view.Therefore,cavity optomechanics has played a significant role in multifarious applications combined with the advantages of diverse physical systems,such as quantum information processing[15,16],weak classical force detection[17],biological sensing[18],precision measurements[19,20],and quantum communication optical storage[21].Meanwhile,some novel phenomena of quantum mechanics have appeared in cavity optomechanics,which has been demonstrated and exploited experimentally,like macroscopic quantum entanglement[22,23],mechanical squeezing[24],normal-mode splitting[25],optomechanically-induced transparency[26,27],or electromagnetically-induced transparency[28],and the nonlinear Kerr effect[29],etc.Furthermore,it is worth noting that a brand-new hybrid system could be constructed by skillfully coupling the cavity optomechanical system with other elements:for example,atom-optomechanical systems[30,31],cavity electromechanical systems[32],a single optical lattice atom[33],and parity-time symmetric systems[34,35].These hybrid systems have become promising candidates for the achievement of quantum manipulation and observation of quantum effects on macroscopic objects in the quantum regime.

For most of the aforementioned potential applications,mechanical resonators should be precooled to their ground-state so that some mechanically mediated quantum phenomena can be observed and explored experimentally.Therefore,the cooling of the mechanical resonator to its quantum ground-state in the cavity optomechanical system is the first area to have been the focus of research.To date,various methods for cooling mechanical resonators have been proposed and demonstrated successively,including pure cryogenic cooling[36],auxiliary cavity cooling[37,38],laser pulse modulations[39],feedback cooling[40–42],and Gaussian pulses[43].A scheme that uses dynamic cavity dissipation to avoid swap heating and accelerate the cooling process has also been proposed[44].A theory of the quantum backaction limit to laser cooling is moved down to zero with a squeezed input light field[45].Furthermore,[46]put forward a scheme to achieve ground-state cooling by periodically modulating the frequencies of the resonator and optical field,cooling down the final mean phonon number below the quantum backaction limit.Meanwhile,a host of concrete implementations about the frequency modulation(FM)of micromechanical resonators and optical components have been reported:for example,the frequency of the cavity mode can periodically change in the time-dependent Jaynes–Cummingstype Hamiltonian model[47],the mechanical resonance frequency can be tuned by electrostatically changing the graphene equilibrium position[48],and other modulation approaches have been realized in superconducting optomechanical systems[49–51].Motivated by these developments,we propose a ground-state cooling scheme of a three-mode optomechanical system,where the frequencies of the two-cavity modes and mechanical mode are modulated.The double-cavity optomechanical system considered in our proposal is a general platform for exploring macroscopic mechanical coherence and quantum information processing.Previously,Gu and Li[52]considered quantum interference effects on an optomechanical cooling system consisting of a two-mode optical cavity.A ground-state cooling scheme via an electromagnetically-induced transparency-like cooling mechanism in a double-cavity optomechanical system has been proposed[53].Liuet al[54]harnessed destructive quantum interference in the all-optical domain of the coupled cavity system for the ground-state cooling of mechanical resonators.The generation of robust optomechanical entanglement induced by the blue-detuning laser and the mechanical gain in a double-cavity optomechanical system has been investigated[55].The coupling channels of the doublecavity system have a complementarity for the decay rates between the two-cavity modes,which can ensure higher cooling efficiency[32].Inspired by methods in[46]and[56],the Stokes heating processes induced by swapping heating and interaction quantum backaction can be fully suppressed via FM.In the current work,we make full use of periodic FM to suppress the Stokes heating processes to realize ground-state cooling more effectively.Here,we use numerical simulations to illustrate the dynamical evolution of a mean phonon number with or without FM in the stable and unstable regions.It is demonstrated that mechanical cooling can be achieved in the resolved-sideband regime,and even in the unresolved-sideband regime,which indicates we can break the mechanical resonators’cooling limit via FM.Therefore,lower and more efficient cooling can be obtained by appropriate adjustment of parameters in the unresolved-sideband regime compared with the single-cavity optomechanical system.

This article is organized as follows.In section 2,the theoretical model of the double-cavity system is described and a linearized Hamiltonian is derived.In section 3,in the absence of FM,the stability conditions for the double-cavity optomechanical system are investigated.To observe the cooling dynamics with and without FM,we illustrate the time evolution of the mean phonon number by calculating the master equation in section 4.Finally,a summary of our work is presented in section 5.

2.Theoretical model and system dynamics

We consider a double-cavity optomechanical system to study the cooling of the mechanical resonator to its ground-state,which is formed by a mechanical mode and two single-cavity modes,as illustrated in figure 1,motivated by[57,58].Here,the mechanical resonator(with frequencyωmand decay rateγm)is coupled to the cavity modea1on the left and the cavity modea2on the other side.Two standard optomechanical subsystems are consequently constructed via the radiation pressure and interaction.At the same time,two monochromatic driving fields(with frequenciesωL1,ωL2)and pumping driving(with amplitudes Ω1,Ω2)are applied to manipulate the optical cavities,respectively.We assume that these two optical cavities have identical characteristics for the sake of simplicity.The full Hamiltonian of the system can be given by(? = 1)

3.Stability conditions

4.Quantum master equation and numerical simulation

4.1.Resolved-sideband cooling

Based on the above calculation results,we are now interested in the discussion of the ground-state cooling of the doublecavity optomechanical system in the resolved-sideband regimeκi<ωm(i=1,2).Here,we choose the red sideband detuningΔ′i=ωm(i=1,2).

4.1.1.Weak coupling regime.In this section,we will study the cooling dynamics of the mechanical resonator in the weak coupling regime∣Gi∣?κi(i=1,2).Firstly,a comparison of the mean phonon number between the current double-cavity optomechanical system and the conventional single-cavity optomechanical system[46]is shown in figure 4.We plot the mean phonon number for different coupling strengthsG2in the case ofG1=0.01ωm.It is evident that the final mean phonon number in the double-cavity system is markedly lower than a single-cavity in weak coupling.In our coupled cavity system,the mechanical resonator can be effectively cooled to the ground state since the Stokes heating processes can be suppressed by the double cooling channel in our current scheme.In figures 5(a)and(b),we compare the time evolution of the mean phonon number with FM or without corresponding to different system parameters.It can show that the cooling effect has not been significantly improved,even with FM in figure 5(a).At this time,the superiority of FM is not obvious due to the fact that the Stokes heating processes themselves have been greatly suppressed in the case of too small cavity decay rates and coupling strengths.However,with the further increase in the cavity decay rates and optical coupling strengths in the same weak coupling region,i.e.

4.1.2.Strong coupling regime.In contrast to the weak coupling regime,the enhancement of cooling processes can be achieved effectively using the current proposed scheme in the strong coupling regimeGi?κi(i=1,2).We show the dynamical evolution of the mean phonon number with different cavity decay rates in figure 6(a).It is noted that the mean phonon number of the system is significantly lower with the addition of FM than without FM.We also find that the final mean phonon number reduces with the increase in cavity decay rates.Furthermore,the system cooling improvement rate will increase with the increase in the cavity decay rates in the same coupling strengths after data verification.For instance,taken as the three sets of parametersthe final mean phonon numberis reduced from 0.261 to 0.206,from 0.147 to 0.102,and from 0.112 to 0.072.The relevant calculation results of the cooling improvement rate are 21.1%,30.6%,and 35.5%,respectively.It can be interpreted that the superiority of the scheme with FM proposed here becomes more prominent in the strong coupling regime.Furthermore,we also plot the time evolution of the mean phonon number with different coupling strengthsG2by introducing FM in the unstable region,as shown in figure 6(b).The cooling of the mechanical resonator fails due to the divergent behavior of the phonon number when the coupling strengths arecorresponding to the unstable region(see the stability conditions in figure 3).However,the final mean phonon number can be cooled to the quantum ground-state successfully by introducing FM.It is evident that,due to the existence of the FM,the Stokes process can be fully suppressed and better cooling of the mechanical resonator can be achieved than without FM,even in the unstable region.

4.2.Unresolved-sideband cooling

We have already discussed the cooling of the resolved-sideband regime in the stable region.In this section,we now explore how to achieve a better cooling effect by optimizing the FM even in the unresolved-sideband regime.Figure 7 shows the dynamical evolution of the mean phonon number with different cavity decay rates inG1=G2=0.2ωm.The cooling effect in the presence of the FM is dramatically improved compared to the conventional double-cavity system.The results indicate that the former cooling limit is much lower than without FM in the stable region.We also note that the mean phonon number cannot be lower than unity for the large cavity decay rate.This is because overlarge cavity decay rates limit the final cooling of the mechanical resonator.Apart from the saturation effect of the cooling rate,the reason for this phenomenon is that the anti-Stokes effect becomes weaker.However,the mean phonon number successfully reaches the ground-state cooling by making use of the FM.This can be interpreted as that FM makes a major contribution to the suppression of the Stokes heating processes,which will be very beneficial to the ground-state cooling experiments of mechanical resonators,even in the unresolvedsideband regime.

Figure 1.A schematic illustration of a double-cavity optomechanical system:a mechanical mode b in the middle is coupled to two optical cavity modes,a1 and a 2 ,which are coupled with each other via phase-dependent phonon-exchange coupling(with the coupling strengths g1 and g2).

Figure 2.The value of the Bessel function of the first kind Jk(ξ)with different normalized modulation amplitudes.

Figure 3.The dynamical stability region for the double-cavity optomechanical system in the absence of FM,in whichG2 is a function ofG ,1 and the other parameters are .

Figure 4.Time evolution of the mean phonon number b with FM between the current double-cavity optomechanical system and the conventional single-cavity optomechanical system is plotted for comparison.Other unmentioned parameters are assumed as .

Figure 5.Time evolution of the mean phonon number with or without FM is plotted for comparison:(a)For (b)ForThe other parameters are selected to be the same as those in figure 4.

Figure 6.(a)Time evolution of the mean phonon number with or without FM for different optical cavity decay by solving the master equation numerically is plotted for comparison:here,(b)Time evolution of the mean phonon numberwith different coupling strengths in the unstable region.The orange dashed line is forand the blue solid line representsThe other parameters are selected to be the same as those in figure 4.

Figure 7.Time evolution of the mean phonon number Nb corresponding to different cavity decay rates with or without FM.Here,G1=G2=0.2 ωm.The other parameters are selected to be the same as those in figure 4.

Figure 8.Time evolution of the mean phonon number Nb for different modulation frequencies by solving the master equation numerically is plotted for comparison.Here,κ1=κ2= 2ωm,G1=G2=0.2ωm.The other parameters are selected to be the same as those in figure 4.

Combining the results of figure 7,we make a reasonable guess that a possible way to achieve a better cooling result is to have a larger frequency of the modulation term in equation(5).In figure 8,the cooling dynamics for different modulation frequencies are depicted with solid curves in different colors.The mechanical resonators have good cooling efficiency when we choose the system parametersIt is noteworthy that the corresponding cooling efficiency is improved as the modulation frequency increases.Nevertheless,the final mean phonon number is almost no different when the modulation frequencies are large enough whenν= 10 andν= 30.The results demonstrate that a too large modulation frequency is confirmed to be of little practical feasibility to the improvement of mechanical cooling results.The Stokes heating process far from the resonant conditions has little effect on the cooling process when the modulation frequency is large enough,or even negligible.It can be explained by a frequency-domain interpretation of optomechanical interactions[46].Therefore,it is particularly crucial to choose a suitable rather than an overlarge modulation frequency to obtain a better cooling effect.

5.Conclusion

In summary,we have used the double-cavity optomechanical system as an example to study the ground-state cooling of a mechanical resonator by solving quantum master equations and employing the covariance approach.In this work,we have theoretically proposed an FM scheme for improving the groundstate cooling of the mechanical resonator from the stable to the unstable region,from the resolved-sideband regime to the unresolved-sideband regime.It is demonstrated by numerical simulations that,in our current scheme,the mechanical resonator can be cooled to its ground-state with a lower limit than in the conventional double-cavity optomechanical system due to the suppression of the Stokes heating processes.In the commonly concerned dynamically stable regime,we have discussed and analyzed the comparison between the time evolutions of the mean phonon number in the weak coupling regime and the strong coupling regime with FM or without.Moreover,in the presence of FM,the final mean phonon number is also well below unity in the unresolved-sideband regime,even if groundstate cooling cannot be achieved in the absence of FM.Finally,we also find that the cooling effect is greatly improved at a higher modulation frequency compared with a lower modulation frequency.The scheme developed in this paper will offer the prospect of the cooling of mechanical resonators,and research and explorations will be richer and more interesting as a result.

Funding

This work was supported by the National Natural Science Foundation of China(Grant No.62061028),the Foundation for Distinguished Young Scientists of Jiangxi Province(Grant No.20162BCB23009),the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics(Grant No.KF202010),the Interdisciplinary Innovation Fund of Nanchang University(Grant No.9166-27060003-YB12),and the Open Research Fund Program of Key Laboratory of Opto-Electronic Information Acquisition and Manipulation of Ministry of Education(Grant No.OEIAM202004).

Appendix.Dynamical equations for the secondorder moments