Entanglement of a Movable Mirror and a Cavity Field by Atomic Coherence

ZHANG Cai-yun, CHANG Yu, PAN Gui-xia, LI Hu, CHEN Hua-jun

(School of Mechanics and Optoelectronics Physics,Anhui University of Science and Technology,Huainan 232001,China)

Abstract：We propose a scheme for Λ-type atoms to entangle an optomechanical oscillator.We show that the larger stationary macroscopic entanglement can be obtained with the existence of atomic medium，and the larger coupling coefficient between the cavity and the atoms corresponds to the larger entanglement.Furthermore,we also find that the entanglement minimum moves toward the side of the big value of Δ/ωm with the larger Rabi frequency,and the entanglement of optomechanical system is insensitive to the cavity losses,which compensates for the effects of the cavity decay.

Key words:optomechanical system;atomic medium;Hamiltonian;entanglement

1 Introduction

Quantum entanglement is a peculiar physical phenomenon in quantum mechanics and has become a pivotal resource of quantum information[1].Experiments have shown that entanglement can occur in microscopic objects such as photons and atoms[1].Therefore,whether entanglement can be generated in mesoscopics or macroscopic objects is a promising subject worth studying[2].

Optomechanical coupling via radiation pressure between mechanical oscillators is beginning to be a promising approach for exploring quantum features at mesoscopic scales[3-8].In recent years,researchers have presented various proposals to generate entanglement of macroscopic objects via radiation pressure coupling.For example,Vitali et al.[9]showed that the entanglement between two movable mirrors can be generated by means of radiation pressure in a classically driven Fabry-Perot cavity.The authors obtained that two separated nanomechanical oscillators can be entangled via radiation pressure by injecting squeezed vacuum light and laser light into the ring cavity[10].Ref.[11] investigated that the steady entanglement of two micromechanical mirrors through two-mode fields can be generated by a correlated emission laser source in a doubly reasonant cavity[11],and explored that the stationary entanglement between an optical cavity field[12]and a macroscopic vibrating mirror can be generated and quantified by means of radiation pressure.

It has recently been shown that the atomic medium has played a crucial role in cavity optomechanical system[13-15].Ref.[13] explored cavity optomechanical coupling assisted by an atomic gas,and showed that the radiation pressure of the cavity field upon the oscillating mirror can be enhanced in the presence of atoms.Genes et al.[13]investigated ground-state cooling of micromechanical oscillator by resonant coupling of two-level ensemble,and Hammerer et al.[15]analyzed a setup to achieve strong coupling between a single trapped atom and a mechanical oscillator.With all previous works shown,the radiation pressure of optomechanical system can be enhanced by injecting atomic medium.

In this paper,we present a method to create a stationary entanglement via atomic coherence.The cavity field and the optomechanical oscillator is entangled when atomic medium is injected into the optomechanical device.We show the steady-state solution of the system and quantify the mechanical entanglement in terms of the logarithmic negativity.

2 Model and steady-state results of the system

Fig.1 Sketch of the system and the atomic configuration.The classical field drive the atomic level |a>?|b> with detuning Δ1while cavity field interact with atomic transition |a>?|c>.

As shown in figure 1,the system under study consists of an optical cavity with a fixed mirror and a movable mirror.Three-level Λ-type atoms is confined in the optical cavity and interacts with the classical laser field.The Hamiltonian of the system is given by

(1)

(2)

whereΔ1=ωa-ωc-ω0,Δ2=ωa-ωb-ω0,δ1=ωa-ωb-ωL,δ2=ωa-ωc-ωL.Considering the effect of damping and noise,the Langevin equations of the hybrid system can be obtained:

(3)

Ps=0,

(4)

(5)

For conrenience,we introduce the quadrature:

(6)

In the rotating-wave approximation,the linear Langevin Equation(5) can be expressed in the following matrix form

(7)

The drift matrixAfor the above equation is given by

The system is stable and reaches its steady state ast→only if all eigenvalues ofAhave negative real parts[9,16].It is difficult to obtain the analytic solution of the eigenvalues of the drift matrix because of high dimensions,so we will guarantee steady state conditions by adopting numerical methods.The covariance matrix can be definedVij()()fj()+fj()fi()〉].When the system is stable,we can get the following Lyapunov equation for the correlation matrix[17].

AV+VAT=-D

(8)

where the noise correlation matrixD=diag[0,γm(2n+1),κ(2n+1),κ(2n+1),0,0,0,0].We can solve the covariance matrix directly.

In order to study the entanglement of the system,we use the logarithmic negativity to quantify the entanglement of the system.The logarithmic negativity is defined as[18]

EN=max{0,-lnη-},

(9)

The system is entangled ifEN>0 or,equivalently,the smallest symplectic eigenvalue of the covariance matrixη-<1/2[19].

We now discuss the entanglement properties of the optomechanical system.In Fig.2,the logarithmic negativityENversus as a function of the normalized detuningΔ/ωmin the presence of the atoms.In Fig.2a,it is noticed that the larger entanglement corresponds to the larger coupling value gi.Fig.2b show that with the increasing of,the entanglement maximum is almost constant,and the minimum moves toward the side of the big value ofΔ/ωm.Fig.2c describes the logarithmic negativityENof the system for different cavity decay rate.It can be observed that the entanglement decreases with the increasingκ.To evaluate the logarithmic negativity between the movable mirror and the cavity mode,we have chosen feasible parameters based on present experiments[20].

Fig.2 The entanglement of a movable mirror and a cavity mode.In(a),the dashed and dash-dotted lines correspond to the parameters g=2×2π×105Hz,and 4×2π×105Hz,κ=2π×2.15×104Hz,bc=2π×105Hz,=2π×6×106.In(b),the dashed and dash-dotted lines correspond to the parametersbc=2π×105Hz and 6π×105Hz,=1π×6×106and 2π×6×106,κ=2π×2.15×104Hz,g=2×2π×105Hz.In(c),the dashed and dash-dotted lines correspond to the parametersκ=2π×2.15×104Hz,and 3π×2.15×104Hz,g=2.0×2π×105Hz,bc=2π×105Hz,=2π×6×106.For all plots,we select the parametersL=5mm,m=20ng,λ=1256nm=10mW,Q′=ωm/γm=6700,a=2×103Hz,Δ1=Δ2=2π×107Hz,and T=42μK.

3 Discussion and Conclusion

It is worth pointing out that we have chosen feasible parameters in present experiments to evaluate the logarithmic negativity between the cavity field and movable mirror.In fact,it is an experiment challenge in entanglement detection concerning entanglement generation between macroscopic mechanical systems.However,it is relatively easy to realize quantum correlation detection in the optomechanical systems.With the advanced technology,we believe that our scheme can be realized.

In conclusion,we present a proposal to generate the stationary entanglement between a fixed mirror and a movable mirror by atomic coherence.With the existence of atomic mediums,the stationary entanglement of the system is numerically studied and quantified in terms of logarithmic negativity.Our study shows that the larger entanglement corresponds to the larger gi,because the entanglement results from the atomic coherence.