Entanglement of Nanomechanical Oscillator anda Cavity Field Enhanced by Atomic Coherence

WANG Xiao-li, PAN Gui-xia, YAO Guan-xin ZHENG Xian-feng, CUI Zhi-feng

(1.Anhui Province Key Laboratory of Optoelectric Materials Science and Technology,Anhui Normal University,Wuhu 241000,China；2.School of Science,Anhui University of Science and Technology,Huainan 232001,China)

Abstract:A scheme for three-level cascade atoms to entangle an optomechanical oscillator and a cavity field is presented.In the presence of atomic mediums,the result shows that the larger stationary macroscopic entanglement of optomechanical system can be obtained and quantified by logarithmic negativity.Moreover,we find that it is insensitive to the cavity decay rate for the entanglement between movable mirror and cavity field,which makes up for the effects of the cavity losses.

Key words:Optomechanical system;Atomic medium;Stationary entanglement;Logarithmic negativity

Introduction

The characteristic trait of entanglement in quantum mechanics has been an attractive topic to study the relevance of the microscopic and macroscopic field[1].Entanglement has been experimentally generated in macroscopic systems,such as irons,atom,and photons[2].However,it is still a challenging task to prepare and measure entanglement in macroscopic bodies due to the decoherence and dissipation[3].

Optomechanical systems are beginning to be promising candidates for studying the macroscopic quantum mechanics[4-7].In recent years,more and more researchers have considerable interest in studying entanglement of macroscopic objects[8-12].Robust entanglement of a micromechanical resonator can be obtained between an optical cavity and a vibrating cavity end-mirror[9].Vitali et al.[11]proposed a scheme to generate stationary entanglement between an optical cavity field mode and a macroscopic vibrating mirror.Refs[12]investigated the entanglement of two separated nanomechanical oscillators in a ring cavity by feeding squeezed light.

The microscopic coherent of atoms plays an important role in cavity electrodynamics system.Refs[13]obtained that the windows analogy electromagnetically induced transparency can be broaden.Chiara et al.[14] investigated the mirror-light entanglement produced by a Bose-Einstein condensate(BEC)in hybrid optomechanial device.Recently,stationary entanglement can be generated in optomechanical systems with the existence of atomic medium[15].

In the present paper,we present a scheme to generate stationary macroscopic entanglement when the atoms are injected into the cavity.The research shows that the larger stationary entanglement can be obtained with the increasingcouplinggi.Moreover,it is insensitive to the cavity dissipation for the entanglement.

1 Model and Steady-State Results of the System

We consider the three-level cascade atoms are injected into the cavity optomechanical system with one fixed partially transmitting mirror and one movable reflecting mirror,as shown in Fig.1.The cavity mode with frequencyωcouple to atomic transition |a>↔|b> and |b>↔|c> and detuningsΔ1andΔ2,respectively.The Hamiltonian of the system can be obtained as

(1)

Fig.1 Sketch of the optomechanical system and the atomic configuration

In interaction picture,the Hamiltonian is written as

(2)

whereδ1=Ea-Eb-ωL,δ2=Eb-Ec-ωL,Δ1=Ea-Eb-ω,Δ2=Eb-Ec-ω.Employing the Hamiltonian (2) and introducing the damping and noise terms into the Heisenberg equations,We can obtain the quantum Langevin equations as

(3)

ħν/κBT)-1]-1.

(4)

The steady-state solutions of the system can be obtained as

(5)

with

(6)

The operators of the system can be written asQ=QS+δQ,P=PS+δP,a=aS+δaby using the linearization approach of quantum optics[12,17]when the cavity is intensely driven (aS≫1).So the linearized Langevin equations are obtained as

(7)

,

2 The Entanglement of Optomechanical Mirror and Cavity Mode

AV+VAT=-D,

(8)

where

D=diag[0,γm(2n+1),0,γn(2n+1),κ(2N+1),κ(2N+1),κ(2N+1),κ(2n+1)].To quantize the stationary entanglement of cavity optomechanical system,we employ the logarithmic negativity EN,which can be obtained as[ 20 ]

EN=max(0,ln2v_)

(9)

Fig.3 The dashed line and dash- dotted and dotted lines correspond to the parametersκ=1.5π×215×103,1.7π×215×103and 2π×215×103Hz,respectively.For all the lines in Fig.3,g=1.5×2π×105Hz and the other parameters are the same with Fig.2.

We give a brief discussion on the experimental feasibility of our scheme.We have chosen feasible parameters based on recent experiment[24].In fact,it is still a challenge to generate entanglement of the system in entanglement detection.However,because of proposed promising programs[11,25],we believe that our scheme can achieve in experiments with the advance of science and technology.

3 Conclusion

In conclusion,we propose a scheme to generate the stationary macroscopic entanglement of optomachanical system by atomic coherence.Our study shows that the larger entanglement corresponds to the larger gi.Furthermore,the largest entanglement decreases with the increasing cavity loss and the others entanglement change little with the existence of atoms.So,in some extent,it compensates the effects of the cavity decay.