Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity
Dong-Yang Wang, Cheng-Hua Bai, Hong-Fu Wang, Ai-Dong Zhu, Shou Zhang
aa r X i v : . [ qu a n t - ph ] D ec Steady-state mechanical squeezing in a hybridatom-optomechanical system with a highly dissipative cavity
Dong-Yang Wang, Cheng-Hua Bai, Hong-Fu Wang ∗ ,
1, 2
Ai-Dong Zhu, and Shou Zhang † Department of Physics, College of Science, Yanbian University,Yanji, Jilin 133002, People’s Republic of China School of Physics, Northeast Normal University,Changchun, Jilin 130024, People’s Republic of China
Quantum squeezing of mechanical resonator is important for studying the macro-scopic quantum effects and the precision metrology of weak forces. Here we give atheoretical study of a hybrid atom-optomechanical system in which the steady-statesqueezing of the mechanical resonator can be generated via the mechanical nonlinear-ity and cavity cooling process. The validity of the scheme is assessed by simulatingthe steady-state variance of the mechanical displacement quadrature numerically.The scheme is robust against dissipation of the optical cavity, and the steady-statesqueezing can be effectively generated in a highly dissipative cavity.
PACS numbers: 42.50.Pq, 07.10.Cm, 42.50.Lc, 42.50.WkKeywords: mechanical squeezing, optomechanical system, dissipation
I. INTRODUCTION
The optomechanical system is a rapidly growing field from the classical Fabry-P´erot in-terferometer by replacing one of the fixed sidewalls with a movable one [1]. The introducedone-dimensional freedom of the movable sidewall can be regarded as a free resonator mode,which can interact with the cavity mode through radiation pressure force originating fromthe light carrying momentum. Many projects of cavity optomechanics systems have beenconceived and experimentally demonstrated in the past decade [2–4]. For example, the ra-diation force has been used for cooling the mechanical resonators to near their quantum ∗ E-mail: [email protected] † E-mail: [email protected] ground states and entangling the cavity and mechanical resonator, and for coherent-statetransiting between the cavity and mechanical resonator [5–14]. Quantum fluctuations be-come the dominant mechanical driving force with strong radiation pressure, which leadsto correlations between the mechanical motion and the quantum fluctuations of the cavityfield [15]. In addition, the optomechanical method of manipulating the quantum fluctua-tions has also been used for generating the squeezing states of the optical and mechanicalmodes [16–19].The history of optical squeezing is linked intimately to quantum-limited displacementsensing [20], and many schemes have been proposed to generate squeezing states in varioussystems [21–23]. The squeezing of light field is proposed for the first time using atomicsodium as a nonlinear medium [22]. In addition, the squeezing of microwave field, whichhas been demonstrated with up to 10 dB of noise suppression [23], is an important tool inquantum information processing with superconducting circuits. In recent years, researchershave found that the optomechanical cavity, which can be regard as a low-noise Kerr non-linear medium [24, 25], can be a better candidate to generate squeezing of the optical andmechanical modes. The squeezing of optical field is easy to be achieved in the optomechan-ical systems, and has been obtained experimentally [16, 26, 27]. However, the squeezing ofmechanical mode has not been observed experimentally. Many schemes have been proposedto generate mechanical squeezing in the optomechanical systems, including methods basedon measurement, feedback, parametric processes, and the concept of quantum-reservoir en-gineering [28–33]. Quantum squeezing of mechanical mode is one of the key macroscopicquantum effects, which can be used for studying the quantum-to-classical transition andimproving the precision of quantum measurements [22, 34–36]. So the mechanical squeezingattracts more and more attentions. For example, in 2011, Liao et al. [17] proposed a schemeto generate mechanical squeezing in a optomechanical cavity. They showed that paramet-ric resonance could be reached approximately by periodically modulating the driving fieldamplitude at a frequency matching the frequency shift of the mirror, leading to an efficientgeneration of squeezing. In 2013, Kronwald et al. [18] proposed a scheme to generate me-chanical squeezing by driving the optomechanical cavity with two controllable lasers withdiffering amplitudes. The scheme utilized a dissipative mechanism with the driven cavityacting as an engineered reservoir. In 2015, L¨u et al. [19] proposed a scheme to generatesteady-state mechanical squeezing via mechanical nonlinearity, which showed that squeez-ing could be achieved by the joint effect of nonlinearity-induced parametric amplificationand cavity cooling process.Traditionally and generally, the decay rate of cavity field, which is a dissipative factor inoptomechanical system, is considered to have negative effect on the performance of quantummanipulation of mechanical modes. Here we propose a method to generate steady-state me-chanical squeezing in a hybrid atom-optomechanical system where the atomic ensemble istrapped in the optical cavity consisting of a fixed mirror and a movable mirror. The coher-ently driving on the cavity mode is a monochromatic laser source which can generate strongoptomechanical coupling between the mechanical and cavity modes. We show that, via themechanical nonlinearity and cavity cooling process in transformed frame, the steady-statemechanical squeezing can be successfully and effectively generated in a highly dissipativecavity.The paper is organized as follows: In Section II, we describe the model of a hybrid atom-optomechanical system and derive the effective coupling between the atomic ensemble andthe mechanical resonator. In Section III, we engineer the mechanical squeezing and derivethe analytical variance of the displacement quadrature of the movable mirror in the steady-state. In Section IV, we study the variance of mechanical mode with the large decay rate ofcavity by numerical simulations method and discuss the validity of the scheme in the highlyand lowly dissipative cavities. A conclusion is given in Section V.
II. SYSTEM AND MODEL
We consider a hybrid atom-optomechanical system depicted in Fig. 1, in which N identicaltwo-level atoms are trapped in the optical cavity consisting of a fixed mirror and a movablemirror. The total Hamiltonian H = H + H I + H pump , which describes the hybrid system,consists of three parts, which reads ( ~ = 1), respectively, H = ω a a † a + ω c S z + ω m b † b + η (cid:0) b + b † (cid:1) ,H I = ¯ g (cid:0) S − a † + S + a (cid:1) − ga † a (cid:0) b + b † (cid:1) ,H pump = Ω d (cid:0) e − iω d t a † + e iω d t a (cid:1) . (1) (cid:15) (cid:71) (cid:71) ω Ω (cid:15) a κ ω (cid:15) (cid:70) (cid:70) γ ω (cid:15) (cid:80) (cid:80) γ ω FIG. 1: (Color online) Schematic diagram of a hybrid atom-optomechanical system with a cloudof identical two-level atoms trapped in an optical cavity consisting of a fixed mirror and a movablemirror. The cavity mode is coherently driven by an input laser with frequency ω d . The part H accounts for the free Hamiltonian of the cavity mode (with frequency ω a anddecay rate κ ), the atoms (with transition frequency ω c and linewidth γ c ), and the mechanicalresonator (with frequency ω m and damping rate γ m ). Here a ( a † ) is the bosonic annihilation(creation) operator of the optical cavity mode, b ( b † ) is the bosonic annihilation (creation)operator of the mechanical mode, and S z = P Ni =1 σ iz is the collective z − spin operator of theatoms. The last term of H describes the cubic nonlinearity of the mechanical resonator withamplitude η . For mechanical resonator in the gigahertz range, the intrinsic nonlinearity isusually very weak with nonlinear amplitude smaller than 10 − ω m . We can obtain a strongnonlinearity through coupling the mechanical mode to an ancillary system [37–40], such asthe nonlinear amplitude of η = 10 − ω m can be obtained when we couple the mechanicalresonator to an external qubit [19].The part H I accounts for the interaction Hamiltonian consisting of the atom-field in-teraction and the optomechanical interaction derived from the radiation pressures. Where¯ g = P Ni =1 g i /N represents the averaged atom-field coupling strength with g i being thecoupling strength between the i th atom and single-photon, and g is the single-photon op-tomechanical coupling strength.The part H pump accounts for the external driving laser with frequency ω d used to co-herently pump the cavity mode. The driving strength Ω d = p P κ/ ( ~ ω d ) is related to theinput laser power P , the mechanical resonator frequency ω d , and the decay rate of cavity κ .The spin operators S − ( S + ) of the atomic ensemble can be transformed to a collectivebosonic operator c ( c † ) in the Holstein-Primakoff representation [7, 21], S − = c p N − c † c ≃ √ N c, S + = c † p N − c † c ≃ √ N c † ,S z = c † c − N , (2)where operators c and c † obey the standard boson commutator [ c, c † ] = 1. Under theconditions of sufficiently large atom number N and weak atom-photon coupling ¯ g , the totalHamiltonian in the frame rotating at input laser frequency ω d is written as H ′ = − δ a a † a − ∆ c c † c + ω m b † b + η (cid:0) b + b † (cid:1) + G (cid:0) a † c + ac † (cid:1) − ga † a (cid:0) b + b † (cid:1) + Ω d (cid:0) a + a † (cid:1) , (3)where δ a = ω d − ω a , ∆ c = ω d − ω c , and G = ¯ g √ N . Applying a displacement transformationto linearize the Hamiltonian, a → α + a, b → β + b, c → ξ + c , where α, β , and ξ are c numbers denoting the steady-state amplitude of the cavity, mechanical, and collectiveatomic modes, which are derived by solving the following equations: h i ( δ a + 2 gβ ) − κ i α − iG ξ − i Ω d = 0 ,ω m β + 3 η (cid:0) β + 1 (cid:1) − g | α | = 0 , (cid:16) i ∆ c − γ c (cid:17) ξ − iG α = 0 . (4)Under the conditions of γ m ≪ κ, γ c , η , the γ m -dependent terms can be neglected. One cansee that when the driving power P is in the microwatt range, the amplitudes of the cavityand mechanical modes satisfy the relationships: | α | , β ≫
1, as shown in Fig. 2. And theamplitudes of the cavity and mechanical modes increase with increasing the driving power.For example, at the point of the driving power P = 2 . × − mW, | α | ≃
160 and β ≃ H L = − ∆ a a † a − ∆ c c † c + ˜ ω m b † b + Λ (cid:0) b + b † (cid:1) + G (cid:0) a † c + ac † (cid:1) − G (cid:0) a + a † (cid:1) (cid:0) b + b † (cid:1) , (5)with ∆ a = δ a + 2 gβ, ˜ ω m = ω m + 2Λ , −3 Driving power P (mW) | α | a nd β | α | β FIG. 2: (Color online) The steady-state amplitudes | α | and β versus the driving power P . Theparameters are chosen to be ω m / (2 π ) = 5 MHz, ω a / (2 π ) = 500 THz, δ a = 2 ω m , ∆ c = ω m , G =0 . ω m , g = 10 − ω m , η = 10 − ω m , κ = 10 ω m , γ c = 0 . ω m , γ m = 10 − ω m , and Ω d = p P κ/ ( ~ ω d ). Λ = 6 ηβ, G = g | α | . (6)And the Hamiltonian of the nonlinear terms, which come from the radiation-pressure inter-action and the cubic nonlinearity, is written as H NL = − ga † a (cid:0) b + b † (cid:1) + (cid:0) ηb † b + ηb † + H . c . (cid:1) . (7)Under the conditions of g, η ≪ Λ , G, G , the nonlinear terms in H NL can be neglectedbecause they are much weaker than the linear terms in H L .Considering the effect of the thermal environment and basing on the linearized Hamilto-nian H L , the quantum Langevin equations for the system are written as˙ a = (cid:16) i ∆ a − κ (cid:17) a − iG c + iG (cid:0) b + b † (cid:1) − √ κa in , ˙ b = (cid:16) − i ˜ ω m − γ m (cid:17) b + iG (cid:0) a + a † (cid:1) − i Λ b † − √ γ m b in , ˙ c = (cid:16) i ∆ c − γ c (cid:17) c − iG a − √ γ c c in , (8)where the corresponding noise operators a in , b in , and c in satisfy correlations h a in ( t ) a † in ( t ′ ) i = h c in ( t ) c † in ( t ′ ) i = δ ( t − t ′ ) , h a † in ( t ) a in ( t ′ ) i = h c † in ( t ) c in ( t ′ ) i = 0 , h b in ( t ) b † in ( t ′ ) i = (¯ n th + 1) δ ( t − t ′ ) , h b † in ( t ) b in ( t ′ ) i = ¯ n th δ ( t − t ′ ), where ¯ n th = { exp [ ~ ω m / ( k B T )] − } − is the mean thermalexcitation number of bath of the movable mirror at temperature T , k B is the Boltzmannconstant, and one recovers a Markovian process. Since the decay rate of cavity, κ , is muchlarger than the linewidth of the atoms, and under the conditions | ∆ a | ≫ | ∆ c | , ˜ ω m ≫ , κ ≫ ( γ c , ω m ) , ω m ≫ γ m , we can approximatively obtain [7] a ( t ) ≃ iG [ b ( t ) + b † ( t )] − i ∆ a + κ − iG c ( t ) − i ∆ a + κ + a (0)exp (cid:16) i ∆ a t − κ t (cid:17) + A ′ in ( t ) , (9)where A ′ in ( t ) denotes the noise term. Neglecting the fast decaying term which containsexp( − κt/
2) and substituting Eq. (9) into Eq. (8), we can obtain the effective couplingbetween the mechanical mode b and collective atoms mode c , which can be written as˙ b = (cid:16) − i ˜ ω ′ m − γ m (cid:17) b + iG eff (cid:0) c + c † (cid:1) − i Λ ′ b † − √ γ m b in , ˙ c = (cid:16) i ∆ eff − γ eff (cid:17) c + iG eff (cid:0) b + b † (cid:1) − √ γ c c in , (10)where the effective parameters of the mechanical frequency, optomechanical couplingstrength, detuning, damping rate, and coefficients of bilinear terms are given by˜ ω ′ m = ˜ ω m + 2 G ∆ a ∆ a + (cid:0) κ (cid:1) ,G eff = (cid:12)(cid:12)(cid:12)(cid:12) GG ∆ a + i κ (cid:12)(cid:12)(cid:12)(cid:12) , ∆ eff = ∆ c − G ∆ a ∆ a + (cid:0) κ (cid:1) ,γ eff = γ c + G κ ∆ a + (cid:0) κ (cid:1) , Λ ′ = Λ + G ∆ a ∆ a + (cid:0) κ (cid:1) . (11)Thus the effective Hamiltonian is rewritten as H eff = − ∆ eff c † c + ˜ ω ′ m b † b − G eff (cid:0) c + c † (cid:1) (cid:0) b + b † (cid:1) + Λ ′ (cid:0) b † + b (cid:1) . (12)When considering the system-reservoir interaction, which results in the dissipations ofthe system, the full dynamics of the effective system is described by the master equation˙ ρ = − i [ H eff , ρ ] + γ eff L [ c ] ρ + γ m (¯ n th + 1) L [ b ] ρ + γ m ¯ n th L [ b † ] ρ, (13)where L [ o ] ρ = oρo † − ( o † oρ + ρo † o ) / γ eff is the effectivedamping rate of the mode c , and ¯ n th is the average phonon number in thermal equilibrium. III. ENGINEERING THE MECHANICAL SQUEEZING
Applying the unitary transformation S ( ζ ) = exp[ ζ ( b + b † ) / ζ = 14 ln (cid:18) ′ ω m (cid:19) , (14)to the total system. Then the transformed effective Hamiltonian becomes H ′ eff = S † ( ζ ) H eff S ( ζ ) = − ∆ eff c † c + ω ′ m b † b − G ′ (cid:0) c + c † (cid:1) (cid:0) b + b † (cid:1) , (15)with ω ′ m = ω m s ′ ω m ,G ′ = G eff (cid:18) ′ ω m (cid:19) − , (16)where ω ′ m is the transformed effective mechanical frequency and G ′ is the transformed ef-fective optomechanical coupling. The transformed Hamiltonian is a standard cavity coolingHamiltonian and the best cooling in the transformed system is at the optimal detuning∆ eff = − ω ′ m . In the transformed frame, the master equation ρ ′ = S † ( ζ ) ρS ( ζ ) of system-reservoir interaction can be approximatively written as [19]˙ ρ ′ = − i h H ′ eff , ρ ′ i + γ eff L [ c ] ρ ′ + γ m (cid:16) ¯ n ′ th + 1 (cid:17) (cid:0) cosh ( ζ ) L [ b ] + sinh ( ζ ) L [ b † ] (cid:1) ρ ′ + γ m ¯ n ′ th (cid:0) cosh ( ζ ) L [ b † ] + sinh ( ζ ) L [ b ] (cid:1) ρ ′ , (17)which is the transformed master equation and can achieve the cooling process. Here ¯ n ′ th =¯ n th cosh(2 ζ )+sinh ( ζ ), is the transformed thermal phonon number. The steady-state densitymatrix ρ can be obtained by solving the master equation Eq. (13) numerically. Defining thedisplacement quadrature X = b + b † for the mechanical mode, the steady-state variance of X is given by h δX i = h X i − h X i , which can be derived as h δX i = (cid:16) n ′ eff + 1 (cid:17) e − ζ , (18)where ¯ n ′ eff is the steady-state phonon number of the transformed system. When the bestcooling (at the optimal detuning ∆ eff = − ω ′ m = − ω m p ′ /ω m ) in the transformedsystem ¯ n ′ eff = 0 is achieved by the cooling process, the steady-state variance h δX i = e − ζ approaches the minimum value. −3.5 −3 −2.5 −2 −1.5 −1 −0.50.60.650.70.750.80.850.90.951 ∆ e ff /ω m V a r i a n ce h δ X i ¯ n th = 1 ¯ n th = 10 ¯ n th = 100 − ω ′ m = − ω m q Λ ′ ω m FIG. 3: (Color online) The variance of the displacement quadrature X relates to the effectivedetuning ∆ eff by solving the master equation numerically. Here ∆ eff can be tuned individually byvarying ∆ c , the average phonon number ¯ n th is set to be 1, 10, and 100 respectively, and the otherparameters are chosen to be the same as in Fig. 2. IV. NUMERICAL SIMULATIONS AND DISCUSSION
In this section, we solve the master equation Eq. (13) numerically to calculate the steady-state variance of the mechanical displacement quadrature X . The relationship between thesteady-state variance and effective detuning is shown in Fig. 3. One can see from Fig. 3 thatthe minimum value of variance can be achieved at the optimal detuning point of ∆ eff = − ω ′ m ,which comes from the standard cavity cooling Hamiltonian in Eq. (15) under the transformedframe. The change rate of variance on the effective detuning increases with increasing theaverage phonon number ¯ n th . In the process of numerical simulation, the parameters areset to be ω m / (2 π ) = 5 MHz, ω a / (2 π ) = 500 THz, δ a = 2 ω m , ∆ c = ω m , G = 0 . ω m , g = 10 − ω m , η = 10 − ω m , κ = 10 ω m , γ c = 0 . ω m , γ m = 10 − ω m , Ω d = p P κ/ ( ~ ω d ), and¯ n th = 1 , ,
100 respectively, which satisfy the conditions | ∆ a | ≫ | ∆ c | , ˜ ω m ≫ , κ ≫ ( γ c , ω m ) , ω m ≫ γ m , ( κ, γ c , η ) ≫ γ m , ( | α | , β ) ≫
1, and (Λ , G, G ) ≫ ( g, η ). Theaverage phonon number ¯ n th = 100 corresponds to the temperature T = 25mK. At theoptimal detuning point ∆ eff = − ω ′ m = − ω m p ′ /ω m , the steady-state variance of thedisplacement quadrature is h δX i = e − ζ = 0 .
64. However, one can see from Fig. 3 thatwe need a more precise control for ∆ eff to achieve the optimal steady-state squeezing of themechanical resonator with the temperature rising constantly.In addition, considering the smaller decay rate of cavity, for example, κ = 0 . ω m , and0 −3 Driving power P (mW) | α | a nd β | α | β FIG. 4: (Color online) The steady-state amplitudes | α | and β versus the driving power P . Theparameters are chosen to be ω m / (2 π ) = 5 MHz, ω a / (2 π ) = 500 THz, δ a = − . ω m , ∆ c = 0 . ω m , G = 0 . ω m , g = 10 − ω m , η = 10 − ω m , κ = 0 . ω m , γ c = 0 . ω m , γ m = 10 − ω m , and Ω d = p P κ/ ( ~ ω d ). −8 −7 −6 −5 −4 −3 −2 −1 00.30.40.50.60.70.80.91 ∆ e ff /ω m V a r i a n ce h δ X i − ω ′ m = − ω m q Λ ′ ω m FIG. 5: (Color online) The variance of the mechanical displacement quadrature X relates to theeffective detuning ∆ eff by solving the master equation numerically. The average phonon numberis set to ¯ n th = 1 and the other parameters are chosen to be the same as in Fig. 4. with the choices of δ a = − . ω m , ∆ c = 0 . ω m , G = 0 . ω m , g = 10 − ω m , η = 10 − ω m , γ c = 0 . ω m , and γ m = 10 − ω m . The relationship between the steady-state amplitudes( | α | , β ) and driving power P is shown in Fig. 4 and the relationship between the steady-statevariance and effective detuning is shown in Fig. 5 (here we calculate the steady-state varianceof the mechanical displacement quadrature X numerically by setting P = 0 . × − mW, | α | = 500, and β = 200), respectively. At the optimal detuning point ∆ eff = − ω ′ m = − ω m p ′ /ω m , the steady-state variance of the displacement quadrature is h δX i = e − ζ = 0 . ζ is achieved at the point of∆ a = κ/
2, which can be easily seen from Eq. (11). Furthermore, the generated steady-statemechanical squeezing in the present scheme can be detected based on the method proposedin Refs. [19, 41]. As illustrated in Refs. [19, 41], for detecting the steady-state mechanicalsqueezing, we can measure the position and the momentum quadratures of the mechanicalresonator via homodyning detection of the output field of another auxiliary cavity modewith an appropriate phase, and the auxiliary cavity is driven by another pump laser fieldunder a much weaker intracavity field so that its backaction on the mechanical mode can beneglected.
V. CONCLUSIONS
In conclusion, we have proposed a scheme for generating the steady-state squeezing ofthe mechanical resonator in a hybrid atom-optomechanical system via the mechanical non-linearity and cavity cooling process in transformed frame. The atomic ensemble is trappedin the optomechanical cavity, which is driven by an external monochrome laser. The effec-tive coupling between the mechanical resonator and the atomic ensemble can be obtainedby reducing the cavity mode in the case of large detuning. We simulate the steady-statevariance of the mechanical displacement quadrature numerically at a determinate laser driv-ing power and find that the steady-state variance has the minimum value at the optimaldetuning point, where the effective detuning is in resonance with the effective transformedmechanical frequency.
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