Quantum entanglement generation and control has been studied in various physical systems^[1] such as electronic,^[2,3] photonic,^[4,5] and atomic and molecular systems,^[6,7] ranging from microscopic systems to mesoscopic devices.^[8] It can help us understand the fundamentals of quantum phenomena, such as quantum decoherence^[9,10] and quantum-classical boundary.^[11] In addition, it has wide applications in quantum information and quantum computation.^[12,13]

Recently, there has been a great deal of interest in optomechanical systems which provides us with a convenient platform to realize macroscopic quantum effects.^[14] Optomechanics explores the coupling between photons and phonons via radiation pressure. Because of the special coupling structure, optomechanics act as a coupling medium for diverse systems.^[15–18] The interaction between photons and phonons can be used in mass sensors,^[19] force and displacement detectors,^[20] and the hardware for realizing quantum information processing.^[12] Long distance photon entanglement can be used to construct quantum communication channels for secure information transmission.^[21] The entanglement between the photons and solid bits,^[22] such as mechanical oscillators and intracavitary atoms can realize the information interaction between flying bits and local bits,^[23,24] as well as quantum information storage.^[13]

Generating and controlling the entanglement of mechanical oscillators are also important subjects in the study of macroscopic quantum effects.^[25,26] It is difficult for us to directly realize the parametric down-conversion (PDC) interaction between two mechanical oscillators in a system containing the interaction Hamiltonian of . To achieve the effective mechanical interaction, the frequency difference between the mechanical oscillators should not be too large. Under this condition, the PDC interaction is very weak in comparison to the beam-splitter (BS) interaction in the dynamic system. In this case, it is hard to generate steady-state entanglement between oscillators. More recently, in order to prepare strongly entangled mechanical modes in the steady state, one can engineer the dissipation of the mechanical modes such that its steady state is the desired target state,^[27–29] and we can also construct bilinear coupling in modulated optomechanics to create steady-state entanglement.^[30] Usually, these schemes need to introduce extra quantum control, which increases the difficulty of the practical implementation of the entanglement proposals.

In this paper, we propose a convenient steady-state entanglement generating and controlling scheme for mechanical oscillators in a composite optomechanical system, in which the movable mirror of the optoemchanical cavity is coupled to a mechanical oscillator. By controlling the adjustable parameters of the classical driving laser, we are able to control the steady-state entanglement between two oscillators. In our analyses, the nonlinear optomechanical interaction, caused by radiation pressure, results in an effective frequency of the movable mirror in the optomechanical system. By choosing appropriate parameters, the effective frequency of the movable mirror can be reversed to −ω_m. In this case, even though the frequency of two oscillators are the same, we can still enhance the PDC interaction and suppress the BS interaction, which generates steady-state entanglement.

Considering an optomechanical system in which the movable mirror is coupled with a mechanical oscillator, as shown in Fig. 1, the Hamiltonian reads ( ) 1 2 3 where a, b₁, and b₂ are the bosonic operators for the optical cavity, movable mirror, and coupled mechanical oscillator with frequencies ω_c, ω_m1, and ω_m2 respectively. The single-photon coupling coefficient of the optomechanical interaction is . H_d describes the standard continuous-wave driving, ω_d is the angular frequency of the laser and ϵ is the cavity driving strength, given by , with P being the input power of the laser and being the input rate of the cavity. In the rotating frame under input laser frequency ω_d, we obtain the quantum Langevin equations, 4 For the purposes of discussion, we split the optical and mechanical field operators into the classical and quantum components: and , where , . The time evolution of the annihilation operators of the system in the Heisenberg picture is then governed by 5 6 where is the detuning between the driving field and the cavity frequency, is the effective detuning of the system. The nonlinear equations (5) dominate the coherent mean-field amplitude evolution of the system which depends on adjustable parameters Δ, κ, and ϵ. These parameters can be controlled by the input driving laser. Therefore, it is possible to control the quantum behavior of the system by utilizing the classical laser field. In blue sideband detuning , the optomechanical system can be used to cool the mechanical oscillator.^[31] In red sideband detuning , the optomechanical system can be used to generate entanglement between photons and phonons.^[1] In strong driving condition , the optical mode of the system can exhibit bistable and chaos behavior.^[32] The nonlinear equations (6) dominate the quantum evolution of the system. The terms and denote the field-enhanced linear interaction between optical mode and mechanical mode b₁. The linearized coupling rate depends on the dynamics of the classical part. The terms and denote the nonlinear interaction between the optical mode and the mechanical mode b₁. In addition, the average displacement of the mechanical oscillator, caused by the radiation pressure, , will lead to a small shift of the detuning frequency . It also adds a term of as an equivalent driving on the coupled mechanical oscillator.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Schematic diagram for the composite optomechanical system, where the optomenchnical oscillator is coupled to an additional mechanical oscillator. This three-mode bosonic system is used to prepare steady-state entanglement between modes b1 and b2. The cavity mode a is driven by a laser with frequency ωd and amplitude ϵ.

Currently, most experimental realizations of cavity optomechanics are still limited in the single-photon weak coupling with a strong driving condition.^[33–35] Under this condition, for the quantum part, we can introduce the so-called “linearized” approximate description^[14] of optomechanics when the nonlinear terms and can be omitted since they are relatively small with respect to the factor α. Thus, equation (6) can be formally integrated as 7

We consider the case of , i.e., the mode is rapidly oscillating in evolution. Then the time evolution of can be approximately described as , where . Substituting it into the equation of , we have 8 where denotes the effective dissipation. Under the condition of , the term containing is a fast decaying term, thus it can be neglected. Then we have 9 We consider the case of , where the second term can be ignored as a higher order small quantity. 10 Substituting into Eqs. (6), we find that the dynamics of modes b₁ and b₂ are independent of mode a. From Eqs. (6) we obtain the approximated expressions, 11 where denotes the effective frequency of the movable mirror, denotes the effective dissipation of the movable mirror, is the effective amplification factor caused by the optomechanical interaction, is the effective damping of the movable mirror. Thus, the effective Hamiltonian is 12 In addition, the mean value β causes an effective driving on the coupled oscillator, i.e. . This equivalent driving is a coherent driving which can be used to enhance the mean number of the coupled oscillator. The effective frequency ω_eff is no longer a fixed parameter. Instead, it can be controlled by the laser driving strength ϵ and detuning Δ. Thus, it is possible to control the quantum behavior between the two oscillators.

As one can tell, the effective frequency ω_eff is a function of G and frequency detuning Δ. Thus, the optical driving field can indirectly modulate the frequency of the movable mirror through the interaction from radiation pressure. When , analogous to the anti-Stokes regime, BS interaction is on resonance in the system, i.e., . This interaction will cause the state transfer between the oscillators, sideband cooling, etc.^[35,36] When , analogous to the Stokes regime, PDC interaction is on resonance in the system, i.e. . This interaction will cause the entanglement of mechanical oscillators, squeeze of mechanical oscillators, etc.^[1,37] Noting that since we select approximately equal to ω_m1, ξ should be large enough to make , so that the effective dissipation will also be magnified. In order to get a large value of ξ and reduce the effective dissipation, a larger G is needed, which means, a strong driving strength ϵ and a small dissipation rate κ are needed.

The interaction between light and mechanical motion due to radiation pressure is intrinsically nonlinear. Several theoretical studies of the nonlinear effect in optomechanical systems have been reported recently.^[13,32,38] Although we have used the linearized approximation in the quantum part, for rigorous consideration, especially for strong driving condition, we should also examine the nonlinear behavior of the optomechanical system when discussing the quantum effect between the two oscillators. Without losing generality, we set in the following discussion. To study the nonlinear behavior of the system, we first consider the classical part of the dynamics, i.e., Eqs. (5). In the steady state they read, 13 Combining the above equations we obtain a third-order polynomial root equation for the mean-field cavity occupation, 14 where is the mean photon number in the cavity, and is the nonlinear parameter. This equation is commonly used to discuss the bistability of the optomechanical system.^[13,32,38] Equation (14) indicates that the mean-field equations have either one or three solutions depending on the number of real roots of the polynomial. As shown in Fig. 2, we exhibit the bistability behavior of the linearized coupling rate G as a function of driving strength ϵ, single-photon coupling rate g, detuning Δ, and cavity dissipation rate κ (the black dashed line denotes the metastable state solution,^[32] the same as in Fig. 3). The same behavior has also been discussed in Ref. [32]. We can obtain a strong linearized coupling rate , in weak single-photon coupling rate and large dissipation rate regime. When using ϵ and G to adjust the value of ω_eff, we need to ensure that the condition is satisfied to ensure the rationality of the approximation. The solution corresponding to the blue-solid line in Figs. 2(a) and 2(b) meets our needs. Therefore, in the following discussion, we will only deal with the steady state solution corresponding to the blue-solid line and ignore others.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) (a) The coupling rate G as a function of driving strength ϵ, where , , , and . (b) G as a function of single-photon coupling rate g, where , , , and . (c) G as a function of detuning frequency Δ, where , , , and . (d) G as a function of dissipation rate κ, where , , , and .

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) (a) and (c) Effective frequency ωeff and effective dissipation rate γeff as a function of driving strength ϵ. The detuning frequency . (b) and (d) ωeff and γeff as a function of detuning frequency Δ. The driving strength . Other parameters are , , and .

The corresponding bistable behavior of the effective frequency ω_eff and the effective dissipation rate are shown in Fig. 3. In Figs. 3(a) and 3(b), we plot the effective frequency ω_eff as a function of ϵ and Δ, showing an obvious bistable behavior of ω_eff. We can adjust the effective frequency from ω_m2 to by increasing driving strength and frequency detuning. In Figs. 3(c) and 3(d), we plot the effective frequency γ_eff as a function of ϵ and Δ, showing an obvious bistable behavior of γ_eff. The effective damping rate increases with the increase of the driving strength, and decreases with the increase of the detuning frequency.

Considering the effect of cavity dissipation rate κ, we plot the density graph of ω_eff as a function of ϵ, κ, and Δ, κ in Figs. 4(a) and 4(b), respectively. The blue-dashed line indicates , and the red-dashed line indicates . As shown in Fig. 4(a), with the increase of the driving strength, the effective frequency will shift from the intrinsic frequency of the oscillator to the negative frequency. It is worth noting that the value of driving strength ϵ = 0 cannot be achieved in our approximation. We draw from ϵ = 0 only for the completeness of the figure. As shown in the blue-dashed line, when the driving strength is large enough, the effective frequency becomes , and the original BS interaction turns into PDC interaction. In addition, to achieve , we need to increase the driving strength with the increase of dissipation rate. Shown as the red-dashed line in Fig. 4(b), when the detuning frequency of the cavity is equal to the oscillator frequency ω_m2, the resonance amplification occurs. Under this condition, the amplification rate ϵ reaches maximum, but the effective frequency is not shifted, which can also be told from the expression of ω_eff. Shown as the blue-dashed line, the effective frequency can reach the value −ω_m under the proper detuning frequency Δ. However, when the dissipation is too large, the optomechanical interaction is not sufficient to make the effective frequency equal to −ω_m. Thus, under the given driving strength, the effective frequency can reach −ω_m with the appropriate detuning frequency. Considering the results in 3 comprehensively, by properly increasing the detuning frequency, the effective frequency ω_eff can achieve −ω_m and the effective dissipation γ_eff can be reduced under the given driving strength. The above analysis shows that, it is possible to adjust the effective frequency from ω_m2 to by controlling the driving strength ϵ and detuning Δ. Especially for the case , the system can be used to generate entanglement between two oscillators.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) Effective frequency ωeff as a function of ϵ, Δ, and κ. (a) ωeff as a function of ϵ and κ, the red-dashed line denotes and the blue-dashed line denotes . The detuning frequency . (b) ωeff as a function of Δ and κ, the red-dashed line denotes and the blue-dashed line denotes . The driving strength . Other parameters are the same as those in Fig. 3.

With the help of the effective Hamiltonian, we investigate the entanglement between the two oscillators. For a nonchaotic dynamical process, the properties of the quantum fluctuations can be determined from a 4 × 4 covariance matrix, , where the elements of the covariance matrix are defined as , , and 15

Thus, the degree of quantum entanglement between the mechanical and optical modes can be assessed by calculating the so-called logarithmic negativity,^[39] defined as , where and . The matrix V_A, V_B, and V_C are 2 × 2 matrices related to the covariance matrix V as 16

We plot the logarithmic negativity E_n as a function of optical driving strength ϵ and frequency detuning Δ with different oscillator coupling strength V in Fig. 5. Note that the linearized optomechanical interaction can lead to a cooling process.^[31] Combining with the phonon–phonon parametric coupling, the system can cool the mechanical modes b₁ and b₂ into an entangled state in the steady state. Therefore, we have the reason to choose the thermal phonon number of the movable mirror to be equal to zero. As shown in Fig. 5(a), for a particular coupling strength V, the peak of entanglement appears when the driving strength is around . Under this driving strength, the effective frequency (this result can be obtained from Fig. 4(a)) which is consistent with the conclusions we have discussed above. It means that we can achieve the purpose of controlling the entanglement of the oscillators by adjusting the driving strength of the cavity. In addition, the degree of the entanglement increases with the oscillator coupling strength V. As shown in Fig. 5(b), for a particular coupling strength V, the peak of entanglement appears at the frequency detuning around . Under this frequency detuning, the effective frequency (this result can be obtained from Fig. 4(b)) which is consistent with the conclusions we have discussed above. This means that we can achieve the purpose of controlling the entanglement of the oscillators by adjusting the frequency detuning of the optical cavity. Again the entanglement increases with the oscillator coupling strength V.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) (a) Entanglement En as a function of ϵ with different coupling V. The detuning frequency . (b) En as a function of ϵ with different coupling V. The driving strength . Other parameters are , , , .

In the present experimental conditions, the mass and scale of the mechanical oscillator can achieve macroscopic level in suspended mirror optomechanical system.^[8,14] The coupling between mechanical resonators can be realized in geometrically interconnected beams^[40] and a charged mechanical resonator via Coulomb interaction.^[41] Therefore, our system has potential applications for the generation and controling of macroscopic entanglement. Under the current experimental conditions, the coherent phonon manipulation in coupled mechanical resonators is difficult to realize. One can utilize two GaAs-based mechanical resonators by using a piezoelectric transducer^[40] or two superconducting lumped-element electrical resonators using an RF SQUID-mediated tunable coupler.^[42] In our scheme, a more convenient way to manipulate the mechanical resonators has been proposed, with which one only needs to control the adjustable parameters of the classical driving laser.

In our result, we proposed a scheme to generate and control the entanglement between two oscillators in a composite optomechanical system. It shows that, under the condition , the radiation pressure from the optical mode can cause an effective frequency ω_eff of the mechanical mirror. The mean value of the movable mirror acts as an effective driving on the coupled oscillator. In addition, this effective frequency can be controlled by the classical optical driving, i.e., driving strength ϵ and frequency detuning Δ. In the case of anti-Stokes regime , the effective frequency of the movable mirror can be adjusted from to −ω_m and the phonon can be cooled to the ground state. Then, we can generate the steady state entanglement between two oscillators by setting the effective frequency , which corresponds to the parametric down-conversion process. Manipulating the effective frequency ω_eff and coupling strength V we can control the strength of the entanglement. Our scheme provides a more convenient way to generate and control the steady state entanglement of mechanical oscillators, which has potential applications in quantum control and fundamental research.