Twofold mechanical squeezing in a cavity optomechanical system
Chang-Sheng Hu, Zhen-Biao Yang, Huaizhi Wu, Yong Li, Shi-Biao Zheng
TTwo-fold mechanical squeezing in a cavity optomechanical system
Chang-Sheng Hu, Zhen-Biao Yang, Huaizhi Wu, Yong Li, and Shi-Biao Zheng Fujian Key Laboratory of Quantum Information and Quantum Optics and Department of Physics,Fuzhou University, Fuzhou 350116, People’s Republic of China Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
We investigate the dynamics of an optomechanical system where a cavity with a movable mir-ror involves a degenerate optical parametric amplifier and is driven by a periodically modulatedlaser field. Our results show that the cooperation between the parametric driving and periodicallymodulated cavity driving results in a two-fold squeezing on the movable cavity mirror that acts asa mechanical oscillator. This allows the fluctuation of the mechanical oscillator in one quadrature(momentum or position) to be reduced to a level that cannot be reached by solely applying eitherof these two drivings. In addition to the fundamental interests, e.g., study of quantum effects at themacroscopic level and exploration of the quantum-to-classical transition, our results have potentialapplications in ultrasensitive sensing of force and motion.
I. INTRODUCTION
Squeezing associated with the mechanical motionof a massive object [1–9] refers to the reduction ofthe quantum fluctuation in its position or momentumbelow the vacuum level, which is not only importantfor fundamental test of quantum theory [10], such asexploration of the quantum-classical boundary [11],but also have potential applications in high-precisionmeasurement [12, 13]. In analogy to the standardparametric techniques applied for squeezing of opti-cal fields, the thermal noise of a mechanical oscillatorcan be reduced directly via parametrical modulationof the mechanical spring constant [14]. However, eventhough the mechanical oscillator is initially preparedin its quantum ground state, the parametric approachfailed to generate a steady-state squeezing of mechan-ical motion below one half of the zero-point level (i.e.the well-known 3-dB limit) due to the onset of insta-bility.In cavity optomechanical systems [15–21], theoret-ical schemes for surpassing the 3-dB limit to realizemechanical squeezing have been proposed, e.g. by in-jecting a broad band squeezed light into the cavityto transfer optical squeezing into mechanical mode[22, 23] or by driving the optical cavity with two-tone control lasers of different amplitudes combinedwith a reservoir engineering technique [24, 25], basedon which the experimental demonstration of station-ary squeezing beyond the 3-dB limit was recentlyachieved [26]. Additionally, it was shown that me-chanical squeezing can also be generated simply by us-ing a periodically amplitude-modulated driving laser[27] or by directly coupling an optical parametric am-plifier (OPA) to the optical cavity [28] , without therequirement of classical feedback and of the input ofsqueezed light [29]. Despite the advantages of eachscheme on certain conditions, it is still highly desirableto further strengthen the mechanical squeezing, andthen the following important problems remain open:Does there exist a cooperative effect when the physicalprocesses used for different methods are applied at thesame time? If yes, to what extent can the mechanicalsqueezing be enhanced by this cooperative effect?In this paper, we study the quantum dynamics of anoptical cavity that has a movable mirror and contains coherent state uncertaintysqueezed state uncertainty
Ω = 2ω (cid:5)
Pumping (cid:6)
OPADriving (cid:7) (cid:8) (cid:9) (cid:10) × × (cid:11) (cid:12) I m (cid:13) R e (cid:13) (b)(a)(c) (d) × × -0.60.40.2-0.4-0.200.4 1.61.0 2.2 Figure 1. (Color online) (a) Sketch of the optomechan-ical setup. The optomechanical cavity that is drivenby a periodically amplitude-modulated laser field con-tains an OPA, which is pumped by a laser of the fre-quency twice of the cavity resonance. See text for de-tails. (b) Two-fold mechanical squeezing. The fluctu-ation in one quadrature of the mechanical mode, re-duced to S by the periodically amplitude-modulatedcavity driving, is further shrunk to S with the addi-tion of the OPA driving. (c), (d) Phase space trajec-tories of the first moments of the mechanical mode for t = [200 τ, τ ] and optical mode for t = [20 τ, τ ] with numerical simulations (blue), and analytical approx-imations of the asymptotic orbits (green dash). Param-eters are ( κ, ∆ , g, γ m ) /ω m = (cid:0) . , . , × − , − (cid:1) , ( E , E +1 , E − ) /ω m = (1 . , . , . × , Λ /κ = 0 . , θ = π , n a = 0 , n m = 100 , and Ω = 2 ω m . a degenerate OPA, and which is driven by a laser fieldwith periodically modulated amplitude, as shown inFig. 1(a). Our results reveal a cooperation-based en-hancement of the squeezing in the fluctuation of themomentum or position of the cavity mirror. Both theparametric pump driving and periodically modulatedcavity driving contribute to the reduction of the me-chanical fluctuation. The resulting two-fold squeezingexceeds the squeezing that can be achieved solely byeither of these two processes [see Fig. 1(b)]. Theidea may be generalized to realize cooperation-based a r X i v : . [ qu a n t - ph ] J un enhancement of other quantum effects in complex op-tomechanical systems, e.g., entanglement between twomechanical oscillators or entanglement between a lightfield and a mechanical oscillator [30, 31]. II. THEORETICAL MODEL
We consider an optomechanical system where a de-generate OPA placed in a Fabry-Perot cavity of length L and finesse F , with one fixed and partially trans-mitting mirror, and one movable and totally reflect-ing mirror [29, 32]. The movable mirror is treated asa quantum-mechanical harmonic oscillator with effec-tive mass m , frequency ω m , and energy decay rate γ m .The cavity mode of resonant frequency ω c is drivenby an external laser of the carrier frequency ω l (alongthe cavity axis) with periodically modulated ampli-tude E ( t ) = (cid:80) + ∞ n = −∞ E n e − in Ω t , where Ω = 2 π/τ with τ > being the modulation period, and the mod-ulation coefficients { E n } are related to the power ofthe associated sidebands { P n } by | E n | = (cid:112) κP n / (cid:126) ω l ,with κ = πc/ (2 F L ) being the cavity decay rate dueto photon leakage through the fixed mirror. The de-generate OPA in the optical cavity is pumped by acoherent field at frequency ω p , which leads to thesqueezing of cavity field [33, 34], affecting the state ofthe movable cavity mirror through the optomechanicalcoupling. We denote the gain of the OPA by Λ (whichdepends on the pumping intensity) and the phase ofthe pump driving as θ . The total Hamiltonian of thesystem in the frame rotating at the laser frequency ω l can be written as ( (cid:126) = 1) H =∆ a † a + ω m p + q ) − ga † aq + i Λ( e iθ a † e − i p t − e − iθ a e i p t )+ i [ E ( t ) a † − E ∗ ( t ) a ] . (1)Here, ∆ = ω c − ω l , ∆ p = ω p − ω l , a and a † are an-nihilation and creation operators of the cavity mode, q and p are the position and momentum operators forthe movable mirror satisfying the standard canonicalcommutation relation [ q, p ] = i , and g = x ZP F ω c /L is the single-photon coupling strength between lightand mechanical oscillator arising from the radiationpressure force, with x ZP F = (cid:112) (cid:126) / mω m being thezero-point motion of the mechanical mode.When the mechanical damping and cavity decay areincluded, the dissipative dynamics of the open sys-tem can be described by the following set of quantumLangevin equations (QLEs) [35] ˙ q = ω m p, ˙ p = − ω m q − γ m p + ga † a + ξ ( t ) , ˙ a = − ( κ + i ∆ ) a + igaq + E ( t ) + 2Λ e iθ a † e − i p t + √ κa in ( t ) , (2)where both the optical ( a in ) and mechanical ( ξ ) noiseoperators have zero-mean value, and the nonzero cor-relation functions of a in are (cid:104) a † in ( t ) a in ( t (cid:48) ) (cid:105) = n a δ ( t − t (cid:48) ) and (cid:104) a in ( t ) a † in ( t (cid:48) ) (cid:105) = ( n a + 1) δ ( t − t (cid:48) ) with n a = [exp( (cid:126) ω c /k B T ) − − being the thermal pho-ton number and that of ξ ( t ) is given by (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = γ m ω m (cid:82) dω π e − iω ( t − t (cid:48) ) ω [1 + coth( (cid:126) ω k B T )] [36, 37]. For thespecific case where the mechanical oscillator has agood quality factor Q ≡ ω m /γ m (cid:29) , ξ ( t ) becomesdelta-correlated (cid:104) ξ ( t ) ξ ( t (cid:48) ) + ξ ( t (cid:48) ) ξ ( t ) (cid:105) / γ m (2 n m +1) δ ( t − t (cid:48) ) [38, 39], which corresponds to the Marko-vian process with n m = [exp( (cid:126) ω m /k B T ) − − beingthe mean thermal excitation number in the mechani-cal mode. III. DYNAMICS OF THE FIRST MOMENTSOF THE OPTICAL AND MECHANICALMODES
Suppose that the external drivings are strongenough such that the intracavity photon number ismuch larger than 1, we can rewrite each Heisenbergoperator as O = (cid:104) O ( t ) (cid:105) + δO ( O = q, p, a ), where δO are quantum fluctuation operators with zero-meanvalues; and justify that (cid:10) a † ( t ) a ( t ) (cid:11) (cid:39) |(cid:104) a ( t ) (cid:105)| and (cid:104) a ( t ) q ( t ) (cid:105) (cid:39) (cid:104) a ( t ) (cid:105) (cid:104) q ( t ) (cid:105) are valid approximations.Applying the standard linearization techniques to theQLEs (2) and setting ∆ p = Ω / for the considerationof mechanical squeezing, we thus obtain the equationsfor the first moments of the optical and mechanicalmodes (cid:104) ˙ q ( t ) (cid:105) = ω m (cid:104) p ( t ) (cid:105) , (cid:104) ˙ p ( t ) (cid:105) = − ω m (cid:104) q ( t ) (cid:105) − γ m (cid:104) p ( t ) (cid:105) + g |(cid:104) a ( t ) (cid:105)| , (cid:104) ˙ a ( t ) (cid:105) = − ( κ + i ∆ ) (cid:104) a ( t ) (cid:105) + ig (cid:104) a ( t ) (cid:105) (cid:104) q ( t ) (cid:105) + E ( t ) + 2Λ e iθ (cid:104) a ( t ) (cid:105) ∗ e − i Ω t , (3)and the linearized QLEs for the quantum fluctuations δ ˙ q = ω m δp,δ ˙ p = − ω m δq − γ m δp + g [ (cid:104) a ( t ) (cid:105) δa † + (cid:104) a ( t ) (cid:105) ∗ δa ] + ξ ( t ) ,δ ˙ a = − ( κ + i ∆) δa + ig (cid:104) a ( t ) (cid:105) δq + 2Λ e iθ δa † e − i Ω t + √ κa in ( t ) , (4)where ∆( t ) = ∆ − g (cid:104) q ( t ) (cid:105) is slightly modulated bythe mechanical motion.The phase space trajectories of the first moments (cid:104) O ( t ) (cid:105) can be found by simulating Eq. (3) for a set oftypical parameters [see Figs. 1(c)-(d)] [40]. When thesystem is far away from the optomechanical instabili-ties and multistabilities [41], the semiclassical dynam-ics in the steady state will evolve toward a fixed orbitwith a period being equal to the modulation periodof the cavity driving τ . Moreover, since the two non-linear terms in Eq. (3) are both proportional to thecoupling strength g , the asymptotic solutions of (cid:104) O ( t ) (cid:105) can then be expanded perturbatively in the powers of g and in terms of the Fourier components for g (cid:28) ω m [27, 42] (cid:104) O ( t ) (cid:105) = ∞ (cid:80) j =0 ∞ (cid:80) n = −∞ O n, j e in Ω t g j . (5)Substituting Eq. (5) into Eq. (3), we can then obtainthe recursive formulas for the time-independent coef-ficients O n, j (see Appendix A). By truncating the se-ries to the first terms with indexes j = 0 , , ..., and n = − , , , we find that the analytical approxima-tions for (cid:104) O ( t ) (cid:105) agree well with the numerical resultsshown in Fig. 1(c)-(d). Thus, the linearized dynamicscan be evaluated with high accuracy for the effectiveoptomechanical coupling simply written as G ( t ) = g + g e − i Ω t + g − e i Ω t , (6)where g n = | g n | e iφ n = √ ∞ (cid:80) j =0 a − n,j g j +1 with n = − , , . IV. QUANTUM FLUCTUATIONS ANDTWO-FOLD MECHANICAL SQUEEZING
To examine the effect of the modulation sidebands( ∼ e ± i Ω t ), we introduce the mechanical annihilationand creation operators δb = ( δq + iδp ) / √ , δb † =( δq − iδp ) / √ . Then, the QLEs for δa and δb are δ ˙ a = − i ∆ δa + iG ( t )( δb † + δb ) + 2Λ e iθ δa † e − i Ω t − κδa + √ κa in ( t ) ,δ ˙ b = − iω m δb − γ m δb − δb † ) + i [ G ( t ) δa † + G ∗ ( t ) δa ]+ √ γ m b in ( t ) , (7)with the mechanical noise operator b in satisfying (cid:104) b in (cid:105) = 0 , (cid:104) b † in ( t ) b in ( t (cid:48) ) (cid:105) = n m δ ( t − t (cid:48) ) , and (cid:104) b in ( t ) b † in ( t (cid:48) ) (cid:105) = ( n m + 1) δ ( t − t (cid:48) ) . We assume thatthe modulation frequency satisfies
Ω = 2 ω m and thecarrier frequency of the laser field driving the cav-ity is close to the anti-Stokes sideband, which leadsto ∆ = ∆ − g (cid:104) q ( t ) (cid:105) (cid:39) ω m for weak optomechanicalsingle-photon coupling. We further assume that thesystem is working in the resolved sideband regime: ω m (cid:29) κ , and the driving fields are weak: ω m (cid:29)| g | , | g − | , | g | . Under these conditions, if we substi-tute the slow varying fluctuation operators δa ( t ) = δ ˜ a ( t ) e − i ∆ t , δb ( t ) = δ ˜ b ( t ) e − iω m t , a in ( t ) = ˜ a in ( t ) e − i ∆ t and b in ( t ) = ˜ b in ( t ) e − iω m t into Eq. (7), the terms ro-tating at ± ω m and ± ω m can be ignored in the ro-tating wave approximation (RWA), which leads to δ ˙˜ a = ig δ ˜ b + ig δ ˜ b † + 2Λ e iθ δ ˜ a † − κδ ˜ a + √ κ ˜ a in ( t ) ,δ ˙˜ b = ig ∗ δ ˜ a + ig δ ˜ a † − γ m δ ˜ b + √ γ m ˜ b in ( t ) . (8)Note that ˜ a in ( ˜ b in ) has the same correlation func-tion as a in ( b in ). We then introduce the opticaland mechanical quadratures with the tilded opera-tors δ ˜ x = ( δ ˜ a + δ ˜ a † ) / √ , δ ˜ y = ( δ ˜ a − δ ˜ a † ) /i √ , δ ˜ q = ( δ ˜ b + δ ˜ b † ) / √ , δ ˜ p = ( δ ˜ b − δ ˜ b † ) /i √ , and thecorresponding noise operators ˜ x in = (˜ a in + ˜ a † in ) / √ , ˜ y in = (˜ a in − ˜ a † in ) /i √ , ˜ q in = (˜ b in + ˜ b † in ) / √ , ˜ p in =(˜ b in − ˜ b † in ) /i √ , in terms of which the QLEs (8) canbe rewritten as ˙ U t ) = ˜
M U ( t ) + N ( t ) , (9)where U ( t ) = [ δ ˜ q, δ ˜ p, δ ˜ x, δ ˜ y ] T , N ( t ) =[ √ γ m ˜ q in , √ γ m ˜ p in , √ κ ˜ x in , √ κ ˜ y in ] , and ˜ M = − γ m g − − Re g − − γ m Re g + Im g + − Im g + − Re g − − κ + 2Λ cos θ
2Λ sin θ Re g + − Im g −
2Λ sin θ − κ −
2Λ cos θ (10)with g ± = g ± g . Note that the stability conditionsderived from the Routh-Hurwitz criterion require theparametric gain to fulfill ¯Λ ≡ /κ < , the calcula-tion of which is fussy and will not be shown here.The mechanical squeezing can be measured by thevariance of the tilded fluctuations (cid:10) δ ˜ q (cid:11) and (cid:10) δ ˜ p (cid:11) ,which are just the first two diagonal elements of thetilded covariance matrix ˜ V i,j ( t ) = [ (cid:104) U i ( t ) U j ( t ) (cid:105) + (cid:104) U j ( t ) U i ( t ) (cid:105) ] / . Using Eqs. (8)-(9), ˜ V ( t ) in the steadystate is dominated by the Lyapunov equation (see Ap-pendix A) ˜ M ˜ V + ˜ V ˜ M T = − D (11)with D = diag [0 , γ m (2 n m + 1) , κ (2 n a + 1) , κ (2 n a + 1)] .Eq. (11) can be analytically solved in the parameterregime with negligible mechanical damping γ m ≈ and null thermal photon number n a = 0 , leading to (cid:10) δ ˜ q (cid:11) = S Ω − − ¯Λ S Λ − , (cid:10) δ ˜ p (cid:11) = S Ω+ + ¯Λ S Λ+ , (12)where S Ω ∓ = ( | g | + | g | ∓ | g || g | cos φ r ) N − , S Λ ∓ = [ | g | cos φ r, + | g | cos φ r, ∓ | g || g | cos( φ r, + φ r, )] N − , N = 2(1 − ¯Λ )( | g | −| g | ) , with φ r ≡ φ − φ , φ r, ≡ θ − φ , and φ r, ≡ θ − φ . Eq. (12) shows that, under theinterplay between the periodic cavity driving andthe parametric interaction, the fluctuations of theposition and momentum of the mechanical oscillatorstrongly depend on the phase matching condition.To clarify the underlying physics clearly, we assume φ r = π, φ r, = π and φ r, = − π , then the variance ofthe position and momentum fluctuations reduce to (cid:10) δ ˜ q (cid:11) = 12 1 + | g g | − | g g | (1 − ¯Λ) − , (13) (cid:10) δ ˜ p (cid:11) = 12 1 − | g g | | g g | (1 + ¯Λ) − , (14)which reveal that the mechanical mode is squeezed inmomentum (i.e. (cid:10) δ ˜ p (cid:11) < . ). Alternatively, the po-sition squeezing can be achieved by setting φ r = 0 ,φ r, = 0 and φ r, = 0 . More importantly, Eq. (14)shows that the cooperation between the two drivingfields results in a two-fold squeezing: The coefficient( − | g g | ) / (1 + | g g | ) describes the squeezing effect pro-duced by the periodically modulated cavity driving,while (1 + ¯Λ) − corresponds to the effect associatedwith the parametric driving.The two-fold mechanical squeezing can be furtherunderstood by introducing the Bogoliubov mode de-fined as δB ≡ δ ˜ b cosh r + e iφ r δ ˜ b † sinh r with tanh r = | g | / | g | [2, 24], which evolves according to the QLEs δ ˙ B = ig ∗ B δ ˜ a − γ m δB + √ γ m B in ,δ ˙˜ a = − κδ ˜ a + ig B δB + 2Λ e iθ δ ˜ a † + √ κ ˜ a in , (15) RWA CRT
Λ(cid:2) = 0.9RWA CRT
Λ(cid:2) = 0.3RWA CRT
Λ(cid:2) = 0RWA CRT
Λ(cid:2) = 0.6 tanh r = 0.2tanh r = 0tanh r = 0.6tanh r = 0.4 C = 0.50 × C = 0.25 × C = 1.00 × C = 0.75 × (cid:3) (cid:4) (cid:5) (cid:6) (cid:3) (cid:4) (cid:5) (cid:6) (cid:3) (cid:4) (cid:5) (cid:6) Λ(cid:2) Λ (cid:2) tanh r Λ(cid:2) (b)(a)(c) (d)
Figure 2. (Color online) (a) (cid:104) δ ˜ p (cid:105) versus cooperativityparameter C under RWA [Eq. (18)] and with counter-rotating terms (CRT) included [Eq. (7)] for, | g | / | g | =0 . , | g − | / | g | = 0 . , and a set of OPA gains ¯Λ . (b) (cid:104) δ ˜ p (cid:105) versus ¯Λ and tanh r for C = 1 . × . The black regionindicates that the mechanical oscillator is not squeezed.The white line denotes the optimal parametric gain ¯Λ withwhich the momentum squeezing reaches its maximum for agiven tanh r . (c) (cid:104) δ ˜ p (cid:105) versus ¯Λ for different cooperativityparameters C with tanh r = 0 . . The black arrows indicatethe optimal squeezing. (d) (cid:104) δ ˜ p (cid:105) versus ¯Λ for differentmodulations of the cavity driving with C = 5 × . Themakers in (c)-(d) indicate the numerical counterpart viathe Fourier transformation, see Appendix C. In all figuresother parameters are κ/ω m = 0 . , n a = 0 , n m = 100 , γ m /ω m = 10 − . with g B = (cid:112) | g | − | g | e iφ . Since the vac-uum state of the Bogoliubov mode corresponds toa squeezed state, the noise input B in for δB haszero mean and the nonzero correlation functions (cid:104) B † in ( t ) B in ( t (cid:48) ) (cid:105) = [( n m + 1) sinh r + n m cosh r ] δ ( t − t (cid:48) ) and (cid:104) B in ( t ) B † in ( t (cid:48) ) (cid:105) = [( n m + 1) cosh r + n m sinh r ] δ ( t − t (cid:48) ) . By applying the adiabatic approx-imation for κ (cid:29) | g B | (i.e. δ ˙˜ a =0) [29], and consider-ing the phase matching condition ( φ r, = π ) for mo-mentum squeezing, we find that the variance of thequadrature δp B = √ i ( δB − δB † ) in the steady statereads (see Appendix B) (cid:104) δp B (cid:105) = κγ m (1 + ¯Λ)4 | g B | (2 sinh r + 1)(2 n m + 1)+ 12(1 + ¯Λ) (2 n a + 1) . (16)For φ r = π , the variance of the momentum fluctu-ation for the original mechanical mode has a simplyanalytical form (cid:104) δ ˜ p (cid:105) = (cosh r − sinh r ) (cid:104) δp B (cid:105) , (17)which is exactly the result of Eq. (14) for γ m = 0 , n a = 0 . This result can be roughly explained as fol-lows: The periodically modulated cavity driving pro-duces a squeezing effect on the momentum fluctuation of the mechanical mode, which mathematically cor-responds to converting the normal mechanical modeinto the Bogoliubov mode through a unitary trans-formation equivalent to a squeezed operator. As aconsequence, the “momentum” fluctuation of the Bo-goliubov mode at the “vacuum” level corresponds tothe normal momentum fluctuation below the vacuumlevel ( (cid:10) δ ˜ p (cid:11) < . ). The parametric driving further re-duces the “momentum” fluctuation of the Bogoliubovmode below the “vacuum” level, resulting in a secondsqueezing effect. V. THE EFFECT OF MECHANICALDAMPING AND EXPERIMENTALFEASIBILITY
Considering the effect of the mechanical damping γ m (cid:54) = 0 , the variances of the fluctuations (cid:10) δ ˜ q (cid:11) and (cid:10) δ ˜ p (cid:11) can again be calculated by the Lyapunov equa-tion (11). As an example, when the phases φ = 0 , φ r = π and θ = π are set and the cooperativityparameter C = 4 | g | / ( κγ m ) is large so that ˜ C ≡ C (1 − tanh r ) (cid:29) , the variance of the mo-mentum is approximately given by (cid:10) δ ˜ p (cid:11) ≈ (cosh r − sinh r ) n m + 1)[ 1 + ¯Λ˜ C + γ m κ (1 + ¯Λ) ] . (18)which agrees well with its numerical counterpart ob-tained by simulation of Eq. (7), as shown in Fig. 2(a).For | g | (cid:39) | g | (corresponding to tanh r → ), the ef-fective coupling between the Bogoliubov mode and thecavity mode becomes negligible, and (cid:10) δ ˜ p (cid:11) (cid:39) n m + is mainly determined by the thermal occupation ofthe mechanical mode with γ m (cid:28) κ , implying that themechanical mode is not squeezed [see Fig. 2(b)] [25].In this case, the self-cooling of the mechanical oscilla-tor through the photon-phonon sideband coupling issuppressed [43–45], therefore, the mechanical oscilla-tor may stay far away from the ground state [24, 46].Generally, there exists an optimal squeezing for (cid:104) δ ˜ p (cid:105) corresponding to the best efficiency of the cooperationbetween the two driving fields, which can be readilyfound by setting d (cid:104) δ ˜ p (cid:105) / d ¯Λ = 0 for an appropriateamplitude modulation | g | / | g | (i.e. a given tanh r ).For ˜ C (cid:29) , (cid:10) δ ˜ p (cid:11) reaches its minimum when the opti-mal parametric gain satisfies ¯Λ opt = η ( (cid:113) ˜ Cη ) − with η = (cosh r − sinh r ) n m + + γ m κ , which is indicated inFig. 2(b). Note that an effective cooperation im-plies a non-negative ¯Λ opt , which imposes a thresholdof ˜ C thr = 4( η − − on ˜ C , namely ˜ C > ˜ C thr . In ad-dition, the stability condition ¯Λ opt < requires ˜ C < ˜ C ins with ˜ C ins = 8(2 η − − , beyond which the bestcooperation efficiency always appears at ¯Λ opt → ,in vicinity of instability, see Fig. 2(c) for the exam-ple of tanh r = 0 . , where we find ˜ C thr ≈ . × ( C thr ≈ . × ) and ˜ C ins ≈ . × ( C ins ≈ ).Considering the set of experimentally feasible pa-rameters [40]: L = 25 mm, F = 1 . × , ω m / π = 1 MHz, Q = 10 , m = 150 ng, T = 5 mK and thepower of the carrier component P = 1 mW of thedriving laser ( λ = 1064 nm), we show in Fig. 2(d) thatthe degree of squeezing for the mechanical momentumwith C = 5 × and thermal occupation n m = 100 is monotonically improved as the dimensionless para-metric gain ¯Λ increases for tanh r = 0 , . , . , . dueto ˜ C > ˜ C ins . Under this condition, the momentumfluctuation is reduced from 0.219 (3.57 dB) to 0.117(6.29 dB) for tanh r = 0 . , and from 0.132 (5.79 dB)to 0.0756 (8.21 dB) for tanh r = 0 . as the parametricgain ¯Λ is increased from 0 to the optimal value 0.99.The momentum squeezing can be further increased fora larger C/n m ratio (the so-called quantum coopera-tivity) under the best efficiency of the two-field coop-eration. Our results clearly show that, with suitablechoice of the system parameters, both the cavity driv-ing and parametric interaction significantly contributeto the reduction of the mechanical momentum fluctu-ation; their cooperation is important for realization ofa strong mechanical squeezing. VI. CONCLUSION
In summary, we have shown that the parametricdriving and the periodically modulated cavity driv-ing, simultaneously applied to a cavity optomechani-cal system, can result in a two-fold squeezing effectson the mechanical oscillator. This enables implemen-tation of strong squeezing for a macroscopic oscilla-tor, which exceeds the result that is solely producedby either of these two drivings. Our results showthat different physical processes, each producing aweak quantum effect, can cooperate to enhance thequantum effect. Our idea can be generalized to morecomplex optomechanical systems to realize two-foldtwo-mode squeezing, offering a possibility to producestrong mechanical-mechanical or optomechanical en-tanglement that can exceed the bound imposed bypresent methods.
ACKNOWLEDGMENTS
This work was supported by the National Natu-ral Science Foundation of China under Grants No.11774058 and No. 11774024, and the Natural Sci-ence Foundation of Fujian Province under Grant No.2017J01401.
Appendix A: Periodic motion and quantumdynamics of the mechanical oscillator in thesteady state
The time-independent coefficients O n, j in theFourier expansion of (cid:104) O ( t ) (cid:105) ( O = q, p, a ) given by Eq.(5) can be found by substituting Eq. (5) into Eq. (3), leading to the following recursive formulas p n, = 0 , q n, = 0 , (A1) a n, = [ κ − i (∆ − ( n + 1)Ω)] E − n + 2Λ e iθ E n +1 [ κ − i (∆ − ( n + 1)Ω)] [ κ + i (∆ + n Ω)] − , corresponding to the zeroth-order perturbation withrespect to g , and with j > , q n,j = ω m j − (cid:88) k =0 ∞ (cid:88) m = −∞ a ∗ m,k a n + m,j − k − ω m − ( n Ω) + iγ m n Ω ,p n,j = in Ω ω m q n,j , (A2) a n,j = 2Λ e iθ a ∗− n − ,j κ + i (∆ + n Ω) + i j − (cid:88) k =0 ∞ (cid:88) m = −∞ a m,k q n − m,j − k − κ + i (∆ + n Ω) . Using Eq. (A1) and (A2), the quantum dynamics ofthe mechanical oscillator can be studied through thelinearized QLEs (4).We introduce the amplitude and phase quadra-tures of the cavity mode as δx = ( δa + δa † ) / √ , δy = ( δa − δa † ) /i √ and the analogous input quan-tum noise quadratures as δx in = ( δa in + δa † in ) / √ , δy in = ( δa in − δa † in ) /i √ for convenience. Then thetime-dependent equations of motion for the quantumfluctuations u ( t ) = [ δq, δp, δx, δy ] T arise as ˙ u ( t ) = M ( t ) u ( t ) + n ( t ) , (A3)with the drift matrix M ( t ) = ω m − ω m − γ m G x ( t ) 2 G y ( t ) − G y ( t ) 0 − κ + 2Λ cos ¯ θ ∆ −
2Λ sin ¯ θ G x ( t ) 0 − ∆ −
2Λ sin ¯ θ − κ −
2Λ cos ¯ θ , and the diffusion n ( t ) = [0 , ξ ( t ) , √ κδx in , √ κδy in ] T being the noise sources. Here ¯ θ = Ω t − θ , and G x , G y are real part and imaginary part of the effectiveoptomechanical coupling G ( t ) ≡ g (cid:104) a ( t ) (cid:105) / √ . If allthe eigenvalues of the matrix M ( t ) have negative realparts at any time (i.e. the Routh-Hurwitz criterion)[47], the system will be in stable in the steady state.On the other hand, since the system in the steadystate will evolve into an asymptotic Gaussian statefor a Gaussian-typed of noise [48], we can then char-acterize the second moments of the quadratures of theasymptotic state through the covariance matrix (CM) V ( t ) , with the matrix elements being V k,l ( t ) = (cid:104) u k ( t ) u † l ( t ) + u † l ( t ) u k ( t ) (cid:105) / . (A4)From Eqs. (A3) and (A4), we can easily derive a lin-ear differential equation governing the evolution of theCM V ( t )˙ V ( t ) = M ( t ) V ( t ) + V ( t ) M T ( t ) + D, (A5)where M ( t ) T is the transpose matrix of M ( t ) , and D = diag [0 , γ m (2 n m + 1) , κ (2 n a + 1) , κ (2 n a + 1)] is a diagonal noise correlations matrix, definedby δ ( t − t (cid:48) ) D k,l = (cid:104) n k ( t ) n † l ( t (cid:48) ) + n † l ( t (cid:48) ) n k ( t ) (cid:105) / .
356 358357 3603593.52.50.501.5 (cid:1) (cid:2)⁄ (cid:4) (cid:5) (cid:6)
Figure 3. (Color online) Variances of the mechanical mo-mentum fluctuation (cid:10) δp ( t ) (cid:11) versus rescaled time t/τ forthe modulation sidebands driving amplitudes E ± and theparametric gain Λ being (i) E ± = 0 , Λ / κ =0.3 (black),(ii) E ± /ω m = 0 . × , Λ /κ = 0 (green), (iii) E ± /ω m =0 . × , Λ / κ =0.3 (red). The dash line represents the mo-mentum variance of the vacuum state (cid:10) δp ( t ) (cid:11) vac = 0 . .Other parameters are the same as in Fig. 1(c),(d). The first two diagonal elements V ( t ) = (cid:10) δq ( t ) (cid:11) , V ( t ) = (cid:10) δp ( t ) (cid:11) of V ( t ) represent the variancesof the fluctuations in the mechanical position andmomentum, and the last two terms V ( t ) = (cid:10) δx ( t ) (cid:11) , V ( t ) = (cid:10) δy ( t ) (cid:11) represent the variances of thefluctuations in the amplitude and phase of the cavitymode. The mechanical oscillator is position- ormomentum-squeezed if either (cid:10) δq ( t ) (cid:11) < / or (cid:10) δp ( t ) (cid:11) < / in the steady state. The degreeof the squeezing can be expressed in the dB unit,which can be calculated by −
10 log (cid:104) δo ( t ) (cid:105) (cid:104) δo ( t ) (cid:105) vac (or o = p, q ), with (cid:10) δq ( t ) (cid:11) vac = (cid:10) δp ( t ) (cid:11) vac = 1 / beingthe position and momentum variances of the vacuumstate.Recalling that the asymptotic behavior of the firstmoments of the mechanical mode and the cavity modeis τ = 2 π/ Ω periodic in the steady state, then we canfind that the drift matrix M ( t ) , which is related to (cid:104) a ( t ) (cid:105) and (cid:104) q ( t ) (cid:105) , satisfies M ( t + τ ) = M ( t ) and there-fore V ( t + τ ) = V ( t ) according to the Floquet the-ory [49]. By solving the evolutional equation (A5) ofthe CM V ( t ) , we have calculated the time-dependentvariances of the mechanical momentum (cid:10) δp ( t ) (cid:11) forthe optomechanical system with (i) OPA, (ii) peri-odic driving, and (iii) both OPA and periodic driv-ing, as shown in Fig.3. It has been realized thatthe cavity solely pumped by parametric interaction(with E ± = 0 ) [29] and solely modulated by pe-riodic driving ( Λ = 0 ) [27] can both lead to me-chanical squeezing, the degree of which (correspond-ing to the minimum of (cid:10) δp ( t ) (cid:11) ) can reach 1.44 dB( (cid:10) δp ( t ) (cid:11) min = 0 . ) for Λ /κ = 0 . , and 5.13 dB( (cid:10) δp ( t ) (cid:11) min = 0 . ) for E ± /ω m = 0 . × , re-spectively. However, we note that, by combining OPAand periodic driving simultaneously, the mechanicalsqueezing will be greatly enhanced, the degree of mo-mentum squeezing can achieve as large as 6.31 dB( (cid:10) δp ( t ) (cid:11) min = 0 . ), which is far beyond the 3 dB limit, required for ultrahigh-precision measurements. Appendix B: The steady-state “momentum”fluctuation of the Bogoliubov mode
The QLEs (15) can be solved in the adiabatic ap-proximation under the condition of κ (cid:29) | g B | . Forthis purpose, we rewrite the equations of motion for δ ˜ a and δ ˜ a † : δ ˙˜ a = − κδ ˜ a + ig B δB + 2Λ e iθ δ ˜ a † + √ κ ˜ a in ,δ ˙˜ a † = − κδ ˜ a † − ig ∗ B δB † + 2Λ e − iθ δ ˜ a + √ κ ˜ a † in . (B1)Setting δ ˙˜ a = δ ˙˜ a † = 0 and θ − φ = 0 , it is readily tofind δ ˜ a = i g B κ (1 − ¯Λ ) ( δB + ¯Λ δB † )+ √ κκ (1 − ¯Λ ) (˜ a in + ¯Λ e iθ ˜ a † in ) . (B2)Inserting Eq. (B2) into δ ˙ B = ig ∗ B δ ˜ a − γ m δB + √ γ m B in , we have δ ˙ B = − | g B | κ (1 − ¯Λ ) ( δB + ¯Λ δB † ) + √ γ m B in + ig ∗ B √ κκ (1 − ¯Λ ) ( δ ˜ a in + ¯Λ e iθ δ ˜ a † in ) . (B3)Here, the term γ m in the coefficient of δB is safely ig-nored in our parameter regime. Then the equation ofmotion for the “momentum” of the Bogoliubov mode δp B is given by δ ˙ p B = − | g B | κ (1 + ¯Λ) δp B + d ( t ) + v ( t ) , (B4)where d ( t ) = i (cid:112) γ m ( B in − B † in ) and v ( t ) = g ∗ B √ κκ (1+¯Λ) (˜ a in − ˜ a † in ) e iθ , whose correlation functions are (cid:104) d ( t ) d ( t (cid:48) ) (cid:105) = γ m (2 sinh r + 1)(2 n m + 1) δ ( t − t (cid:48) ) and (cid:104) v ( t ) v ( t (cid:48) ) (cid:105) = | g B | κ (1+¯Λ) (2 n a + 1) δ ( t − t (cid:48) ) , respectively.Based on these equations, we obtain the equation for (cid:10) δp B (cid:11) as ∂ (cid:104) δp B (cid:105) ∂t = − | g B | κ (1 + ¯Λ) (cid:104) δp B (cid:105) + | g B | κ (1 + ¯Λ) (2 n a + 1)+ γ m r + 1)(2 n m + 1) . (B5)As a result, the steady-state “momentum” fluctua-tion of the Bogoliubov mode can be solved by setting ∂ (cid:104) δp B (cid:105) ∂t = 0 , giving rise to Eq. (16) in the main text. Appendix C: Optomechanical squeezing in therotating wave approximation
Taking the Fourier transform of Eq. (9) byusing f ( t ) = π (cid:82) + ∞−∞ f ( ω ) e − iωt dω and f † ( t ) = π (cid:82) + ∞−∞ f † ( − ω ) e − iωt dω , we obtain the position andmomentum fluctuations of the movable mirror in thefrequency domain, i.e. δ ˜ q ( ω ) = A ( ω )˜ x in ( ω ) + B ( ω )˜ y in + E ( ω )˜ q in ( ω ) + F ( ω )˜ p in ,δ ˜ p ( ω ) = A ( ω )˜ x in ( ω ) + B ( ω )˜ y in + E ( ω )˜ q in ( ω ) + F ( ω )˜ p in , (C1)where A ( ω ) = − √ κid ( ω ) { [Λ( α − α ) + iu ( ω )Im( g − g )] v ( ω )+ i ( | g | − | g | )Im( g − g ) } ,B ( ω ) = √ κd ( ω ) { [Λ( β − β ) − u ( ω )Re( g − g )] v ( ω ) − ( | g | − | g | )Re( g − g ) } ,E ( ω ) = √ γ m d ( ω ) { [ u ( ω ) − ] v ( ω )+( | g | − | g | ) u ( ω ) + Λ(Γ − Γ ) } ,F ( ω ) = √ γ m d ( ω ) i Λ[( g − g ) e − iθ − ( g ∗ − g ∗ ) e iθ ] ,A ( ω ) = √ κd ( ω ) { [Λ( β + β ) + u ( ω )Re( g + g )] v ( ω )+( | g | − | g | )Re( g + g ) } ,B ( ω ) = √ κid ( ω ) { [Λ( α + α ) − iu ( ω )Im( g + g )] v ( ω ) − i ( | g | − | g | )Im( g + g ) } ,E ( ω ) = √ γ m d ( ω ) i Λ[( g + g ) e − iθ − ( g ∗ + g ∗ ) e iθ ] ,F ( ω ) = √ γ m d ( ω ) { [ u ( ω ) − ] v ( ω )+( | g | − | g | ) u ( ω ) − Λ(Γ − Γ ) } , (C2)with α = g e − iθ − g ∗ e iθ , α = g e − iθ − g ∗ e iθ , β = g e − iθ + g ∗ e iθ , β = g e − iθ + g ∗ e iθ , Γ = g e − iθ + g ∗ e iθ , Γ = g e − iθ + g ∗ e iθ , u ( ω ) = γ m − iω , v ( ω ) = κ − iω and d ( ω ) = [ u ( ω ) v ( ω ) + ( | g | − | g | )] − v ( ω ) . (C3)The first two terms in δ ˜ q ( ω ) and δ ˜ p ( ω ) originatefrom the radiation pressure contribution, and the lasttwo terms are from the thermal noise contribution. Without optomechanical coupling ( g = g = 0 ),the mechanical mode subjected to the purely thermalnoise will make quantum Brownian motion leading to δ ˜ q ( ω ) = √ γ mγm − iω ˜ q in and δ ˜ p ( ω ) = √ γ mγm − iω ˜ p in . The expres-sions of the spectra for the position and momentumfluctuations of the mechanical mode are ( Z = ˜ q, ˜ p ) S Z ( ω ) = π (cid:82) + ∞−∞ dω (cid:48) e − i ( ω (cid:48) + ω ) t × [ (cid:104) δZ ( ω ) δZ ( ω (cid:48) ) (cid:105) + (cid:104) δZ ( ω (cid:48) ) δZ ( ω ) (cid:105) ] , (C4)which can be solved by using the correlation functionsof the noise sources in the frequency domain [29] (cid:104) ˜ q in ( ω )˜ q in ( ω (cid:48) ) (cid:105) = (cid:104) ˜ p in ( ω )˜ p in ( ω (cid:48) ) (cid:105) = ( n m + )2 πδ ( ω + ω (cid:48) ) , (cid:104) ˜ q in ( ω )˜ p in ( ω (cid:48) ) (cid:105) = − (cid:104) ˜ p in ( ω )˜ q in ( ω (cid:48) ) (cid:105) = i πδ ( ω + ω (cid:48) ) , (cid:104) ˜ x in ( ω )˜ x in ( ω (cid:48) ) (cid:105) = (cid:104) ˜ y in ( ω )˜ y in ( ω (cid:48) ) (cid:105) = ( n a + )2 πδ ( ω + ω (cid:48) ) , (cid:104) ˜ x in ( ω )˜ y in ( ω (cid:48) ) (cid:105) = − (cid:104) ˜ y in ( ω )˜ x in ( ω (cid:48) ) (cid:105) = i πδ ( ω + ω (cid:48) ) , (C5)and are given by S ˜ q ( ω ) = [ A ( ω ) A ( − ω ) + B ( ω ) B ( − ω )]( n a + )+[ E ( ω ) E ( − ω ) + F ( ω ) F ( − ω )]( n m + ) ,S ˜ p ( ω ) = [ A ( ω ) A ( − ω ) + B ( ω ) B ( − ω )]( n a + )+[ E ( ω ) E ( − ω ) + F ( ω ) F ( − ω )]( n m + ) , (C6)where the first term proportional to ( n a + ) and thesecond term proportional to ( n m + ) correspond tothe radiation pressure contribution and thermal noisecontribution, respectively. 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