Cavity-induced mirror-mirror entanglement in a single-atom Raman laser
CCavity-induced mirror-mirror entanglement in single-atom Raman laser
Berihu Teklu,
1, 2, ∗ Tim Byrnes,
2, 3, 4, 5, 6 and Faisal Shah Khan Quantum Computing Research Group, Applied Math and Sciences, Khalifa University, Abu Dhabi, UAE New York University Shanghai, 1555 Century Avenue, Pudong, Shanghai 200122, China State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences,East China Normal University, Shanghai 200062, China NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan Department of Physics, New York University, New York, NY 10003, USA (Dated: May 22, 2019)We address an experimental scheme to analyze the optical bistability and the entanglement of two movablemirrors coupled to a two-mode laser inside a doubly resonant cavity. With this aim we investigate the masterequations of the atom-cavity subsystem in conjuction with the quantum Langevin equations that describe theinteraction of the mirror-cavity. The parametric amplification-type coupling induced by the two-photon coher-ence on the optical bistabilty of the intracavity mean photon numbers is found and investigated. Under thiscondition, the optical intensities exhibit bistability for all large values of cavity laser detuning. We also providenumerical evidence for the generation of strong entanglement between the movable mirrors and show that it isrobust against environmental thermalization.
PACS numbers: 42.50.Wk, 07.10.Cm, 42.50.Ex, 85.85.+j
I. INTRODUCTION
Optomechanics explores the interaction between light andmechanical objects, with a strong experimental focus onmicro- and nanoscale systems. It has potential to observequantum effects like entanglement on a macroscopic objectand to apply them to quantum information processing. Fur-thermore, it may provide a new paradigm for quantum metrol-ogy, precision measurement and non-linear dynamical sys-tems. Quantum effects in optomechanical systems (OMS) ledto the demonstration that non-classical states can be gener-ated in an optical cavity [1, 2]. Moreover, the entanglementbetween a cavity mode and a mechanical oscillator has beenstudied both in the steady state [3, 4] and in the time domain[5, 6]. In the case of hybrid OMS, the bipartite entanglementbetween an atomic ensemble, cavity modes and a mirror [7]has shown that a strongly coupled system showing robust tri-partite entanglement which can be realized in continuous vari-able (CV) quantum interfaces [8].Several schemes have been proposed to generate entangle-ment between a pair of oscillators interacting with a commonbath or in a two-cavity OMS [9, 10]. The long-lived entangle-ment between two membranes inside a cavity has been stud-ied [11] for two mirrors coupled to a cavity [12–15]. Severalother works have used atomic coherence to induce entangle-ment [2, 12, 14]. At this point, we should note that entan-glement in microcavities is by no means restricted to Ramanlasers. As a matter of fact, the formalism can be extended,e.g. to the intersubband case [16], where a strong interplaybetween photons and the cavity can play an important role forboth fundamental and applied physics in the THz to mid In-frared range [17–20]. ∗ Electronic address: [email protected]
There have been several quantum features of OMS investi-gating the generation of macroscopic entangled states for cav-ity fields due to the atomic coherence in a two-mode laser [21–25], e.g. the generation of two-mode entangled radiation in athree-level cascade atomic medium [21, 25], a four-level sin-gle atom [22], and a four-level Raman-driven quantum-beatlaser [23]. Entanglement of nanomechanical oscillators andtwo-mode fields has been achieved via radiation pressure cou-pling in a cascade configuration due to microscopic atomiccoherence. In Ref. [12], two macroscopic mirrors were entan-gled via microscopic atomic coherence injected into the cavityand Ref. [14] proposed an additional scheme is proposed forentangling two-mode fields whose entanglement can be trans-ferred to two movable mirrors through radiation pressure in acontrolled emission laser. The two photon coherence is gener-ated by strong external classical fields which couples the samelevels (dipole-forbidden) in the cascade scheme.In this paper, we consider a scheme for entanglement gen-eration of two micromechanical mirrors in a four-level atomsin an N configuration through two-mode fields generated bya correlated laser source in a doubly resonant cavity. All thetransitions of interest are dipole allowed. We take the initialstate of a four-level atom to be a coherent superposition be-tween of one of the two lower and upper atomic levels, re-spectively. We show that, in contrast to previous work to theusual the S -shaped bistability observed in single-mode op-tomechanics [26–28], our scheme shows that the optical in-tensities of the two cavity modes exhibit bistabilities for allvalues of detuning, due to the parametric amplification-typecoupling induced by the two-photon coherence. We also stud-ied the entanglement created between two movable mirrors inthe adiabatic regime and our scheme can control the degree ofentanglement with an external field driving the gain medium.This paper is organized as follows. The scheme and theHamiltonian of the system are introduced in Sec. II A. In Sec.II B we analyze the bistability and entanglement between the a r X i v : . [ qu a n t - ph ] M a y (b) 𝜔𝜔 𝐿𝐿 𝜔𝜔 𝐿𝐿 𝜔𝜔 𝑝𝑝 𝜔𝜔 𝑐𝑐 (a) 𝜈𝜈 𝜈𝜈 FIG. 1: (a) Schematics of a two-mode laser coupled to two movablemirrors M and M . The doubly-resonant cavity is driven by twoexternal lasers of frequency ω L and ω L , and the cavity modes, fil-tered by a beam splitter (BS), are coupled to their respective movablemirrors by radiation pressure. (b) The gain medium is a single Ra-man atom. Two external laser drives of frequencies ω p and ω are alsoapplied to generate two-photon coherence. movable mirrors, by means of a master equation for the two-mode laser coupled to thermal reservoirs. In Sec II C we de-rive the linearized quantum Langevin equations for the field-mirror subsystem. In Sec. III we study the coupling inducedby the two photon coherence on the bistability of the mean in-tracavity photon numbers for both cases (RWA and BRWA).We present a method in Sec. IV to study the entanglementgeneration between two mirrors using a full numerical analy-sis of the system. Concluding remarks are given in Sec. V. II. MODEL OF THE SYSTEMA. Hamiltonian
The OMS which we consider consists of a Fabry-P´erot cav-ity of length L with two movable mirrors driven by two modecoherent fields as shown in Fig. 1(a). The laser system is con-sists of a gain medium of four level atoms in a N configurationas shown in Fig. 1(b). We take the initial state of a four-levelatom to be in a coherent superposition of either the state be-tween | a (cid:105) and | c (cid:105) or | d (cid:105) and | b (cid:105) . Moreover, a driven laser withamplitude Ω p and frequency ω p couple the levels | d (cid:105) and | c (cid:105) and another laser with amplitude Ω and frequency ω couple the levels | a (cid:105) and | b (cid:105) . The atoms are injected into the doublyresonant cavity at a rate r a and removed after time τ , whichis longer than the spontaneous emission time. For the purposeof this paper we take the initial state between | a (cid:105) and | c (cid:105) . Thetwo cavity modes with frequencies ν and ν interact nonres-onantly with each atom. We consider the movable mirrors as aquantum harmonic oscillators, so the system can be describedby the following Hamiltonian ( (cid:126) = 1 ) H = (cid:88) j = a,b,c,d ω j | j (cid:105)(cid:104) j | + (cid:88) j =1 ν j a † j a j + g ( a † | b (cid:105)(cid:104) d | + a | d (cid:105)(cid:104) b | ) + g ( a † | c (cid:105)(cid:104) a | + a | a (cid:105)(cid:104) c | )+ Ω p ( | d (cid:105)(cid:104) c | e − iω p t + H.c. ) + Ω( | a (cid:105)(cid:104) b | e − iωt + H.c. )+ (cid:88) j =1 [ ω m j b † j b j + G j a † j a j ( b j + b † j )]+ i (cid:88) j =1 ( ε j a † j e − iω Lj t − H.c. ) (1)where ω j is the frequency of the j th level, ν j is the frequencyof the j th cavity mode, g j is the atom-field coupling, Ω p and Ω are the amplitudes of the drive lasers that couple the | c (cid:105) → | d (cid:105) and | b (cid:105) → | a (cid:105) transitions respectively, and ω p , ω are the frequencies of the drive lasers. ω m j are the frequen-cies of the movable mirrors, b j ( b † j ) are the annihilation (cre-ation) operators for the mechanical modes and the relation G j = ( ν i /L j ) (cid:112) (cid:126) /m j ω m j is the optomechanical couplingstrength, | ε j | = (cid:112) κ j P j / (cid:126) ω L j is the amplitude of the exter-nal pump field that drive the doubly resonant cavity, with κ j , P j , and ω L j being the cavity decay rate related with outgo-ing modes, the external pump field power and the frequenciesof the pump laser, respectively. In (1), the first two terms de-note the free energy of the atom and the cavity modes, thethird and the fourth terms represent the atom-cavity mode in-teraction, the fifth and sixth terms describe the coupling ofthe levels | d (cid:105) ↔ | c (cid:105) and | a (cid:105) ↔ | b (cid:105) by the drive laser and thelast three terms describe the free energy of mechanical oscilla-tors, the coupling of the mirrors with the cavity modes and thecoupling of the external laser derives with the cavity modes,respectively.Using the fact that (cid:80) j | j (cid:105)(cid:104) j | = 1 , the Hamiltonian (1) canbe rewritten (after dropping the unimportant constant ω b ) as H = H + H I where H = ω | a (cid:105)(cid:104) a | + ( ω − ν ) | c (cid:105)(cid:104) c | + ν | d (cid:105)(cid:104) d | + ν a † a + ν a † a (2)Now applying the transformation exp( iH t ) H I exp( − iH t ) ,we obtain the Hamiltonian in the interaction picture as the sumof the following terms V = ∆ c | a (cid:105)(cid:104) a | + (∆ c − ∆ ) | c (cid:105)(cid:104) c | + ∆ | d (cid:105)(cid:104) d | + g ( a † | b (cid:105)(cid:104) d | + a | d (cid:105)(cid:104) b | ) + g ( a † | c (cid:105)(cid:104) a | + a | a (cid:105)(cid:104) c | )+ Ω p ( | d (cid:105)(cid:104) c | + H.c. ) + Ω( | a (cid:105)(cid:104) b | + H.c. ) (3) V = (cid:88) j =1 [ ω m j b † j b j + G j a † j a j ( b j + b † j )]+ i (cid:88) j =1 ( ε j a † j e iδ j t − H.c. ) (4)where we have assumed for simplicity ν + ν = ω p + ω and we define δ j = ν j − ω L j . The master equation for thelaser system can be derived using the terms that involve theatomic states V following the standard laser theory methods[29]. In order to obtain the reduced master equation for thecavity modes, it is convenient to trace out the atomic states. B. Master equation
In order to obtain the dynamics of the system we require themaster equation corresponding to the Hamiltonian (3). Ourmodel for this system is similar to many earlier treatments ofa two-mode three-level laser [25, 29, 30]. Following theseworks, we give a derivation of the master equation for ourcase (proof can be found in the Appendix V), here we justshow the main result. Assuming that the atoms are injectedinto the cavity at a rate r a , we can write the density matrixfor the atomic and cavity system ρ AR at time t as the sum ofthe density operator of the cavity modes plus a single atominjected at an earlier time multiplied by the total number ofatoms in the cavity for an interval ∆ t . Taking the continuumlimit, assuming that the atoms are uncorrelated with the elec-tromagnetic modes at the time the atoms are injected into thecavity and when they are removed, and tracing out the atomicdegrees of freedom, we obtain the following equation for thedensity matrix of the cavity modes: ddt ρ ( t ) = − ig (cid:16) a † ρ db + a ρ bd − ρ bd a − ρ db a † (cid:17) − ig (cid:16) a ρ ca + a † ρ ac − ρ ac a † − ρ ca a (cid:17) + κ L [ a ] ρ + κ L [ a ] ρ (5)The last terms in (5) are the Lindblad dissipation terms [31],where κ j are the cavity damping rates added to account for thecoupling of the cavity modes with thermal Markovian reser-voirs. The conditional density operators ρ ac = (cid:104) a | ρ AR | c (cid:105) and ρ db = (cid:104) d | ρ AR | b (cid:105) can be obtained from the master equation ofthe atomic and cavity system, resulting in ddt ρ ac ( t ) = − ( γ ac + i ∆ ) ρ ac − ig ( a ρ cc − ρ aa a ) − i (Ω ρ bc − Ω p ρ ad ) (6) ddt ρ bd ( t ) = − ( γ bd − i ∆ ) ρ bd + ig ( ρ bb a † − a † ρ dd ) − i (Ω ρ ad − Ω p ρ bc ) (7)where γ ac and γ bd are the dephasing rates for the off-diagonaldensity matrix elements. We make use of the linear approxi-mation by keeping terms only up to second order in the cou-pling constants g j in the master equation. This is justifiedbecause the coupling constants of the two quantum fields aresmall as compared to other system parameters [ ? ]. Afterobtaining the zeroth-order dynamical equations for the con-ditional density operators other than ρ ac and ρ bd , the good-cavity limit is applied, where the cavity damping rate is muchsmaller than the dephasing and spontaneous emission rates.The cavity variables then vary more slowly than the atomicones. The atomic variables reach the steady state earlier thanthe cavity ones, so we can set the time derivatives of the afore-mentioned conditioned density operators to zero, being able tosolve the system of equations analytically (see Appendix V).Here we consider the case in which the atoms are injectedinto the cavity in a coherent superposition of the levels | a (cid:105) and | c (cid:105) . We take the initial state as | Ψ A (0) (cid:105) = C a (0) | a (cid:105) + C c (0) | c (cid:105) , so the initial density operator for a single atom hasthe form ρ A (0) = | Ψ (cid:105) A (cid:104) Ψ | A = ρ (0) aa | a (cid:105) (cid:104) a | + ρ (0) cc | c (cid:105) (cid:104) c | +( ρ (0) ac | a (cid:105) (cid:104) c | + H.c ) , (8)where ρ (0) aa = | C a | , ρ (0) cc = | C c | , and ρ (0) ca = C c C ∗ a arethe initial two level atomic coherence, which can lead tosqueezing and entanglement accompanying light amplifica-tion [21, 22, 32–35]. It is convenient to introduce the quan-tity η ∈ [ − , to parametrize the initial density matrix as ρ (0) aa = − η , and we set the initial coherence to ρ ac (0) = (1 − η ) / . The equations of motion, (6) and (7) can besolved using the adiabatic approximation, so that the follow-ing equations are obtained: − ig ρ bd = ξ a ρ − ξ ρa + ξ ρa † − ξ a † ρ (9) − ig ρ ac = − ξ a ρ + ξ ρa − ξ ρa † + ξ a † ρ (10)The explicit expressions for the coefficients ξ i can be found inAppendix V. Finally, the master equation for the cavity modestakes the form ddt ρ ( t ) = ξ ( a † ρa − a a † ρ ) + ξ ∗ ( a † ρa − ρa a † )+ ξ ( a ρa † − ρa † a ) + ξ ∗ ( a ρa † − a † a ρ )+ ξ ( a ρa † − a † a ρ ) + ξ ∗ ( a ρa † − ρa † a )+ ξ ( a † ρa − ρa a † ) + ξ ∗ ( a † ρa − a a † ρ )+ ξ ( ρa a − a ρa ) + ξ ∗ ( a † a † ρ − a † ρa † )+ ξ ( a a ρ − a ρa ) + ξ ∗ ( ρa † a † − a † ρa † )+ ξ ( ρa † a † − a † ρa † ) + ξ ∗ ( a a ρ − a ρa )+ ξ ( a † a † ρ − a † ρa † ) + ξ ∗ ( ρa a − a ρa )+ 12 (cid:88) i =1 κ i [( N i + 1)(2 a i ρa † i − a † i a i ρ − ρa † i a i )+ N i (2 a † i ρa i − a i a † i ρ − ρa i a † i )] , (11)where we have included the damping of the cavity modes bytwo independent thermal reservoirs with mean photon number N i . C. Heisenberg-Langevin formulation
In order to study the entanglement between the two mov-able mirrors, we need the quantum Langevin equations forthe cavity modes and the mechanical system. Including thecreation and annihilation operators for the mechanical sys-tem in the Hamiltonian V and making use of the expression (cid:68) ˙ O (cid:69) = T r ( O ˙ ρ ) we obtain ˙ a = − (cid:18) κ − ξ (cid:19) a + ξ a † − iG a ( b † + b ) + ε e iδ t + F (12) ˙ a = − (cid:18) κ ξ (cid:19) a − ξ a † − iG a ( b † + b ) + ε e iδ t + F (13) ˙ b j = − iω m j b j − γ m j b j − iG j a † j a j + √ γ m j f j (14)where ξ = ξ ∗ − ξ ∗ , ξ = ξ ∗ − ξ ∗ , ξ = ξ − ξ and ξ = ξ − ξ . The quantum noise operators F and F ap-pear as a result of the coupling of the external vacuum with thecavity modes and through spontaneous emission. The terms f j are the noise operators corresponding with a thermal reser-voir coupled to the mechanical oscillators.The quantum noise operators F µν have zero mean andsecond-order correlations given by (cid:104) F µ ( t ) F ν ( t (cid:48) ) (cid:105) = 2 D µν δ ( t − t (cid:48) ) . (15)Using the generalized Einstein’s relations [36, 37] (cid:104) D µν (cid:105) = −(cid:104) A µ ( t ) D ν ( t ) (cid:105)−(cid:104) D µ ( t ) A ν ( t ) (cid:105) + ddt (cid:104) A µ ( t ) A ν ( t ) (cid:105) (16) where we find that the only nonzero noise correlations be-tween F , F , F † and F † are (cid:104) F † ( t ) F ( t (cid:48) ) (cid:105) = 2[ Re ( ξ ) + κ N ] δ ( t − t (cid:48) ) , (17) (cid:104) F ( t ) F † ( t (cid:48) ) (cid:105) = 2[ Re ( ξ ) + N ( N + 1)] δ ( t − t (cid:48) ) , (18) (cid:104) F † ( t ) F ( t (cid:48) ) (cid:105) = 2[ Re ( ξ ) + κ N ] δ ( t − t (cid:48) ) , (19) (cid:104) F ( t ) F † ( t (cid:48) ) (cid:105) = 2[ Re ( ξ ) + κ ( N + 1)] δ ( t − t (cid:48) ) , (20) (cid:104) F ( t ) F ( t (cid:48) ) (cid:105) = [ ξ ∗ + ξ ] δ ( t − t (cid:48) ) . (21)Meanwhile, f j are the noise operators with zero mean con-tributed by mechanical oscillators and fully characterized bytheir correlation functions (cid:104) f † j ( t ) f j ( t (cid:48) ) (cid:105) = n j δ ( t − t (cid:48) ) , (22) (cid:104) f j ( t ) f † j ( t (cid:48) ) (cid:105) = ( n j + 1) δ ( t − t (cid:48) ) , (23)where n j = [exp( (cid:126) ω m j /κ B T j ) − − , is the mean thermaloccupation number and κ B represents the Boltzmann con-stant, and T j is describing the temperature of the reservoirof the mechanical resonator. III. BISTABILITY OF INTRACAVITY MEAN PHOTONNUMBERSA. Mean field expansion
Typically the single-photon coupling is very weak, butthe optomechanical interaction can be greatly enhanced byemploying a coherently driven cavity. Bistability has beenobserved in driven cavity optomechanical systems using aFabry-P´erot-type optomechanical system in the optical do-main [27, 38]. In this section we proceed to study the effectof the coupling induced by the two photon coherence on thebistability of the mean intracavity photon numbers. In orderto understand the bistability from the perspective of the intra-cavity photon number, we consider the steady-state solutionsof (12)-(14). This can be performed by transforming the cav-ity field to its rotating frame, defined by ˜ a j = a j e − iδ j t , andexpanding operators around their mean value: ˜ a j = (cid:104) ˜ a j (cid:105) + δ ˜ a j ˜ b j = (cid:104) ˜ b j (cid:105) + δ ˜ b j . (24)Here, (cid:104) ˜ a j (cid:105) is the average cavity field produced by the laser de-rive (in the absence of optomechanical coupling), and δ ˜ a j rep-resents the quantum fluctuations around the mean (assumed tobe small). We have also neglected the highly oscillating terms exp[ − i ( σ + σ ) t ] in the transformed frame that contains boththe fluctuations δ ˜ a j and classical mean values (cid:104) ˜ a j (cid:105) . In orderto obtain the solutions for (cid:104) ˜ a j (cid:105) in the steady state, one musteither simplify the equations by making a rotating wave ap-proximation which neglects the fast oscillating terms, or solvea set of self-consistent equations. We show both of these ap-proaches in the following subsections. B. Rotating wave approximation
In the Rotating wave approximation (RWA), we neglect fastoscillating terms in the transformed quantum Langevin ap-proach to determine the evolution for (cid:104) ˜ b j (cid:105) and (cid:104) ˜ a j (cid:105) . This givesthe steady-state solutions according to (cid:104) b † j + b j (cid:105) = − ω m j G j I j γ m j / ω m j , (25) (cid:104) ˜ a j (cid:105) = ε j iδ j + κ j / − j η j , (26)where I j = |(cid:104) ˜ a j (cid:105)| (27)is the steady-state intracavity mean photon number, δ j = ν j − ω L j + G j (cid:104) b † j + b j (cid:105) (28)is the cavity mode detuning, and we have chosen the fre-quency shift due to radiation pressure δν j ≡ G j (cid:104) b † j + b j (cid:105) (29)for convenience. We have also defined η = ξ ∗ − ξ ∗ η = ξ − ξ . (30)We can then write the equations for the intracavity mean pho-ton numbers to have the implicit form I j (cid:12)(cid:12)(cid:12)(cid:12) i ( δ j − β j I j ) + κ j − j η j (cid:12)(cid:12)(cid:12)(cid:12) = | ε j | , (31)where we have used δ j = ν j − ω L j (32)and β j = (2 ω m j G j ) / ( γ m j / ω m j ) . (33)Eq. (31) is of the form of the standard equations for S -shapedbistabilities for intracavity intensities in an optomechanicalsystem with effective cavity damping rates κ j + +2( − j η j .We would note that typically in RWA, there is no couplingbetween the intensities of the cavity modes that is due to thetwo-photon coherence induced in the system.To demonstrate the bistable behavior of the mean intracav-ity photon numbers in doubly resonant cavity, we use a set ofparticular parameters from recent available experimental se-tups [39, 40]. We consider mass of the mirrors m = 145ng ,cavity with lengths L = 112 µ m , L = 88 . µ m , pump laserwavelengths λ = 810nm , λ = 1024nm , rate of injec-tion of atoms r a = 1 . , mechanical oscillator dampingrates γ m = γ m = 2 π × , mechanical frequencies ω m = ω m = 2 π × , and without loss of generality,we assume that the dephasing and spontaneous emission rates ( pW ) I ⨯ δ = π MHz δ = π MHz δ = FIG. 2: Tunable optical bistability of intracavity field. The leftpanel shows the phase diagram for the intra-cavity mean photonnumber I for different values of cavity laser detuning δ and ex-ternal pump field strength (cavity drive laser) P in the rotating waveapproximation. The right panel red, green, and blue curves are thecross section of the phase diagram for cavity laser detuning δ =6 π MHz , π MHz , , respectively. Here g = g = 2 π × , Ω /γ = 10 , κ = κ = 2 π × , and assuming that all atomsare initially in their excited state | ψ A (0) (cid:105) = | c (cid:105) corresponding to theparameter η = 1 . for the atoms γ ac = γ bd = γ cd = γ ab = γ bc = γ ad = γ a = γ b = γ c = γ d = γ = 3 . . For the purpose of this pa-per, we assume a Gaussian distribution for the atom densityand set both one- and two-photon detunings to , ∆ p = 0 and ∆ c = 0 , respectively.We first illustrate the bistability of the steady-state intracav-ity mean photon number for the first cavity mode. The first ex-ample we present in Fig. 2 is the steady-state intracavity pho-ton number under the red-detuned ( δ > ) frequency range.We point out that we have introduced the effective detuningfor our system (32), where the red-detuned regime occurs forall positives values of the effective detuning, which is the op-posite to prior conventions [41]. The left panel of Fig. (2)shows that the optical bistability regime persists for a broaderrange of the external pump fields. The right panel shows thatan S-shaped behavior of the bistable intra-cavity mean photonnumber for I . The strength of the bistability is changed byincreasing the intensity of the external field and the detuning.For the second cavity mode, we have found almost exactlythe same results for the bistability behavior of the steady-stateintracavity mean photon number. C. Beyond the rotating wave approximation
Let us analyze the bistability behavior of the intracavitymean photon number in the NRWA. In this case we are ableto see the effect of the two-photon coherence. To study thebistabilty in the regime, we consider the rotating frame de-fined by the bare cavity frequencies ν j . This is equivalent tothe assumption that the cavity mode detunings δν j = 0 in theHamiltonian (4). It stays in the counter-rotating terms in theLangevin equations for ˜ a j . This approach can be traced backthe condition δ = − δ ≡ − δ . (34)The expectation values of the cavity mode operators with thischoice of detuning are (cid:104) ˜ a (cid:105) = ε α ∗ + ε ( ξ ∗ − ξ ∗ ) α α ∗ + ( ξ ∗ − ξ ∗ )( ξ ∗ − ξ ∗ ) (35) (cid:104) ˜ a (cid:105) = ε α ∗ − ε ( ξ − ξ ) α ∗ α + ( ξ − ξ )( ξ − ξ ) , (36)where α = i ( δ − β I ) + κ / − η α = − i ( δ + β I ) + κ / η . (37)As can be seen in (35) and (36), the coupling between (cid:104) ˜ a (cid:105) and (cid:104) ˜ a (cid:105) is due to the coefficients ξ and ξ , which are pro-portional to the coherence induced either by the coupling ofatomic levels by an external laser, or by injecting the atoms ina coherent superposition of upper and lower levels. Here, weconsider a more general expression by introducing a new pa-rameter that relates the cavity drive amplitudes ( P ∼ µ P ) | ε | = µ | ε | ≡ µ | ε | . (38)We thus obtain an equivalent relation for the intracavity meanphoton number | α ( I ) α ∗ ( I ) + ( ξ ∗ − ξ ∗ )( ξ ∗ − ξ ∗ ) | | α ∗ ( I ) + µ ( ξ ∗ − ξ ∗ ) | I = | ε | , (39) | α ∗ ( I ) α ( I ) + ( ξ − ξ )( ξ − ξ ) | | µα ∗ ( I ) − ( ξ − ξ ) | I = | ε | . (40)The above transformation provides an elegant approach tounderstanding the effect of the coupling on the bistability be-havior of the cavity modes by examining the limits of the pa-rameter µ . In the limit where µ (cid:28) ( P (cid:28) P ), the de-nominator in (40) can be approximated as | µα ∗ − ( ξ − ξ ) | ≈ | µ ( − iδ + κ / − ( ξ − ξ )) − ( ξ − ξ ) | (41)for µ β I / | ( ξ − ξ ) | (cid:28) . In this case, the ratio of (39)and (40) yields a cubic equation I /I = | α ∗ ( I ) + µ ( ξ ∗ − ξ ∗ ) | | µ ( − iδ + κ / − ( ξ ∗ − ξ ∗ )) − ( ξ − ξ ) | . (42)Note that this implies that I exhibits bistability when the in-tensity of the first cavity mode is varied.An exact numerical analysis on Eqs. (39) and (40) is shownin Fig. (3), which indicates that the behavior of the cavitymode mean photon number is very sensitive to the sign ofdetuning. As can be seen in the RWA case, the bistabilty oc-curs in the “red detuned” regime ( δ > )-good agreement isachieved in the regimes of validity of each model. We also ob-serve that the bistable region widens with increasing detuningand derive laser power. ( pW ) ⨯ - I ⨯ δ = π MHz ( pW ) ⨯ - I ⨯ δ = π MHz ( pW ) ⨯ - I ⨯ δ = π MHz ( pW ) ⨯ - I ⨯ δ = π MHz
FIG. 3: Cross section of the phase diagram at δ / π = 6MHz , δ / π = 3MHz , δ / π = 1MHz , and δ / π = 0 . . No-tice that the bistability appears for positive values of detuning, whichgood agreement is achieved in the “red detuned” regime which al-lows in single-mode optomechanics [26–28]. Here we have used µ = 0 . ( P = 0 . P ), and atoms are initially injected into thecavity in state | ψ A (0) (cid:105) = | c (cid:105) , that is, for the value of the parameter η = 1 . See text and Fig. 2 for the other parameters. IV. DYNAMICS OF CONTINUOUS VARIABLEENTANGLEMENT
In this section, we investigate the degree of entanglement ofthe movable mirrors of the doubly resonant cavity in the adi-abatic regime. The detection of entanglement in similar con-texts has been investigated by many groups recently [21, 32–34, 42–44]. Although there is no entanglement between thecavity fields and the movable mirrors, here we will show thatthe entanglement between the two-mode fields can be trans-ferred to entanglement between the movable mirrors of thedoubly resonant cavity. Indeed, optimal entanglement trans-fer from the two-mode cavity field to the mechanical modesis achieved by eliminating adiabatically the dynamics of thefield modes, specifically in circumstances where κ j (cid:29) γ m j .We introduce the slowly varying fluctuation operators δa j ≡ δ ˜ a j e iδ j t and ˜ b j ≡ b j e iω mj t and using (12)-(14), thecorresponding linear quantum Langevin equations are written δ ˙ a = − κ (cid:48) δa + ξ δa † − iG (cid:104) ˜ a (cid:105) ( δ ˜ b † e i ( δ + ω ) t + δ ˜ b e i ( δ − ω ) t ) + F (43) δ ˙ a = − κ (cid:48) δa − ξ δa † − iG (cid:104) ˜ a (cid:105) ( δ ˜ b † e i ( δ + ω ) t + δ ˜ b e i ( δ − ω ) t ) + F (44) δ ˙˜ b j = − γ m j δ ˜ b j − iG j (cid:104) ˜ a j (cid:105) δa † j e i ( ω mj + δ j ) t − iG j (cid:68) ˜ a † j (cid:69) δa j e i ( ω mj − δ j ) t + √ γ m j f j (45)where κ (cid:48) = κ − ξ , κ (cid:48) = κ + 2 ξ . Here we havethe choice of using using the RWA when evaluating (cid:104) ˜ a j (cid:105) . Inthe RWA, the model should not enter the regime where themeasurement is capable of resolving the zero-point motion ofthe oscillator in a time short compared with the mechanicaloscillation period. This regime – which requires very strongoptomechanical coupling – exhibits interesting behavior, in-cluding dynamical mechanical squeezing [45, 46]. From theperspective of quantum state transfer, it has been shown in[28, 47] that the optomechanical interaction and consequentlythe field-mirror entanglement are enhanced when the detun-ing of each cavity-driving field is δ j = − ω m j . To avoidthese issues, we explicitly compute the (cid:104) ˜ a j (cid:105) without usingthe RWA by using a self-consistent iterative approach. Set-ting δ j = − ω m j and using the adiabatic approximation forthe δa j equations we get the expressions for the mirror vari-ables. Moreover, we can choose the phase of the driving laserin such a way that (cid:104) ˜ a j (cid:105) = − i |(cid:104) ˜ a j (cid:105)| . Hence, we have the finalexpressions δ ˙˜ b = − γ m δ ˜ b + α (cid:0) e iδ t − e − iδ t (cid:1) δ ˜ b + α (cid:16) − e − i ( δ + δ ) t (cid:17) δ ˜ b † + (cid:101) F , (46) δ ˙˜ b = − γ m δ ˜ b + α (cid:0) e − iδ t − e iδ t (cid:1) δ ˜ b + α (cid:16) e − i ( δ + δ ) t − (cid:17) δ ˜ b † + (cid:101) F (47)where α ≡ ξ κ G G |(cid:104) ˜ a (cid:105)| |(cid:104) ˜ a (cid:105)| (cid:101) F ≡ κ (cid:48) κ G |(cid:104) ˜ a (cid:105)| (cid:16) e − iδ t F − F † (cid:17) + 4 ξ κ G |(cid:104) ˜ a (cid:105)| (cid:16) e − iδ t F † − F (cid:17) + √ γ m f α ≡ ξ κ G G |(cid:104) ˜ a (cid:105)| |(cid:104) ˜ a (cid:105)| (cid:101) F ≡ κ (cid:48) κ G |(cid:104) ˜ a (cid:105)| (cid:16) e − iδ t F − F † (cid:17) + 4 ξ κ G |(cid:104) ˜ a (cid:105)| (cid:16) F − e − iδ t F † (cid:17) + √ γ m f (48)with κ (cid:48) = κ − ξ , κ (cid:48) = κ +2 ξ and κ = κ (cid:48) κ (cid:48) +4 ξ ξ .In order to study the entanglement between the mirrors itis convenient to define the position and momentum operators as δq j = δ ˜ b j + δ ˜ b † j √ and δp j = i δ ˜ b † j − δ ˜ b j √ . Once we get the ex-pressions for these fluctuation operators we can write a matrixequation of the form ˙ u ( t ) = M ( t ) u ( t ) + n ( t ) , (49)where we define u = (cid:0) δq , δp , δq , δp (cid:1) T . (50)Here M is a matrix containing the coupling between the fluc-tuations and the vector n contains the noise operators of boththe cavity and the mirrors. This inhomogeneous differentialequation can be solved numerically. The evolution of thequadrature fluctuations is described by the general solution of(49) is formally expressed as [5, 6, 48] u ( t ) = G ( t ) u (0) + G ( t ) (cid:90) t G − ( τ ) n ( τ ) dτ, (51)where G ( t ) = e (cid:82) M ( s ) ds (52)and the initial condition satisfies G (0) = I , where I is theidentity matrix. To bring quantum effects to the macroscopiclevel, one important way is the creation of entanglement be-tween the optical mode and the mechanical mode. If the initialstate of the system is Gaussian, the statistics remain Gaussianunder continuous linear measurement for all time.The entanglement can therefore be quantified via the log-arithmic negativity. The logarithmic negativity is a conve-nient and commonly used parameter to quantify the strengthof a given entanglement resource and has the attractive proper-ties of both being additive for multiple independent entangledstates and quantifying the maximum distillable entanglement[49]. Here, we will quantify the entanglement by means ofthe logarithmic negativity. In particular, such measurementcan be obtained from the correlation matrix V with elementsgiven by V i,j ≡ (cid:104) u i u j + u j u i (cid:105) + (cid:104) u i (cid:105) (cid:104) u j (cid:105) (53)fully characterizes the mechanical and optical variances (Italso includes information on the quantum correlation betweenthe two mechanical and the optical cavity modes), giving riseto a block structure: V = (cid:18) A CC T B (cid:19) (54)The corresponding logarithmic negativity E N is given by [50,51] E N = max (cid:0) , − ln 2 η − (cid:1) , (55)where η − = 1 √ (cid:113) Σ − (cid:112) Σ − det V (56)is the symplectic eigenvalue with regards to quantum correla-tions and Σ = det A + det B − C . (57)The interesting quantities in the present model are the quadra-ture fluctuations of the cavity and the mirror. Since the fluctu-ations are time dependent, so will be the elements of the cor-relation matrix. In order to compute its elements, we define acovariance matrix R ( t ) by the elements R (cid:96),(cid:96) (cid:48) ( t ) = (cid:104) u (cid:96) ( t ) u (cid:96) (cid:48) ( t ) (cid:105) (58)for (cid:96), (cid:96) (cid:48) = 1 , , , .In order to quantify the two-mode entanglement, we needto determine the covariance matrix R ( t ) . Taking into accountthe (49) and assuming that the correlation between its ele-ments and the noise operators at the initial state is zero, thegeneral expression for the covariance matrix R ( t ) at an arbi-trary time has takes the form R ( t ) = G ( t ) R (0) G T ( t ) + G ( t ) Z ( t ) G T ( t ) , (59)where Z ( t ) = (cid:90) t (cid:90) t G − ( τ ) C ( τ, τ (cid:48) ) (cid:2) G − ( τ (cid:48) ) (cid:3) T dτ dτ (cid:48) . (60)The elements of the matrix C ( τ, τ (cid:48) ) are the correlation be-tween the elements of the vector n , that is, C l,m ( τ, τ (cid:48) ) = (cid:104) n l ( τ ) n m ( τ (cid:48) ) (cid:105) . (61)Those elements can be easily calculated by using the general-ized Einstein relation for the noise operators. Moreover, sincethe expectation value for the noise operators is zero, the equa-tion for the mean value of the fluctuations is (cid:104) u ( t ) (cid:105) = G ( t ) (cid:104) u (0) (cid:105) . (62)The above are the formal equations for the evolution ofquadrature operators of the mirrors. We now assume the den-sity matrix of the initial conditions of the mirrors is separa-ble and the mechanical bath is, as usual, in a thermal state attemperature T with occupancy n th and the cavity mode is invacuum state. Therefore, the initial density matrix for the i thmechanical oscillator is given by ρ m i = ∞ (cid:88) j =0 n ji (1 + n i ) j +1 | j (cid:105) (cid:104) j | . (63)Under this assumption the R matrix at the initial state is givenby R (0) = n + i − i n + n + i − i n + . (64) Ω = Ω = Ω = t E N FIG. 4: Logarithmic negativity E N of the micromechanical mirrorsfor the cavity drive lasers’ for thermal phonon numbers, n = n =50 and thermal photon numbers N = N = 1 as a function of t at constant Ω p /γ = 0 . with Ω = 15 (solid green line),
Ω = 20 (dotted red line), and
Ω = 30 (dotdashed black line). Here g =g = 2 π × , κ = κ = 2 π × , and assuming thatall atoms are initially in their excited state | ψ A (0) (cid:105) = | c (cid:105) , that is, forthe value of the parameter η = 1 . From (55), the entanglement of the movable mirrors can beeasily computed numerically.In Fig. 4, we plot the degree of entanglement of the twomovable mirrors tunable as a function of time t for differ-ent Ω at fixed input laser powers P , thermal noises n andthermal photon numbers N . We consider the standard casewhere the case of symmetric mechanical damping ( γ = γ = γ ), symmetric thermal occupation of the mechanical baths( n = n = n ) and symmetric thermal photon numbers( N = N = N ). This allows fast numerical results forthe time dependent second moments to be evaluated. The as-sumption of equal thermal occupations is reasonable for mostexperimental situations, while it turns out that our results arenot sensitive to unequal mechanical damping rates providedthat they are both small. We observe that the amount of en-tanglement decreases with time and it is clear that the amountof entanglement are the same provided that with increasingthe external driving field, Ω and saturated for all values ofthe external driving field. In turn this analysis shows that thegenerated entanglement can be controlled by adjusting exper-imental conditions, particularly the external driving field Ω .To see the effect of the cavity-driving laser powers P on theoutput entanglement, we plot the time dependence of the en-tanglement for various cavity-driving laser powers when allatoms are initially in their excited state | ψ A (0) (cid:105) = | c (cid:105) , thatis, for the value of the parameter η = 1 in Fig. 5 for thermalphonon numbers n = n = n = 5 and thermal photon num-bers N = N = N = 1 . We observe that the degree ofentanglement E N increases and persists for longer time whenthe cavity-driving power P decreases and the two movablemirrors are entangled for a wide range of the drive laser pow-ers and saturated (for P < . µ W). This is due to the cou-pling of the cavity-field mode to a mirror. We next examine P = = μ WP = t E N FIG. 5: Logarithmic negativity E N of the micromechanical mirrorsfor thermal phonon numbers n = n = 5 and thermal photon num-bers N = N = 1 as a function of t at constant Ω p /γ = 0 . and Ω /γ = 5 with P = P = P = 0 . nW (solid red line),P = P = P = 0 . µ W (dotted blue line), and P = P = P = 0 . mW (solid green line). Here g = g = 2 π × , κ = κ = 2 π × , and assuming that all atoms are ini-tially in their excited state | ψ A (0) (cid:105) = | c (cid:105) , i.e., for the value of theparameter η = 1 . the effect of the thermal noise on the degree of entanglement.The degree of entanglement of the two movable mirrors, as afunction of time with the external driving field held constant,are shown in Fig. 6. We observe in the figures that the de-gree of entanglement has a similar curve to the effects of thecavity drive lasers for a small input power P and a small ther-mal noise n . We also see that the degree of entanglement forthe mirrors is reduced with increasing temperature. We seethat the critical time above which the logarithmic entangle-ment E N disappears increases with decreasing phonon ther-mal numbers. This is reminiscent of entanglement sudden-death where it does not exponentially decay but goes to zeroat a critical time [52, 53].Finally, we address the environmental temperature depen-dence of the two movable mirrors, as shown in Fig. 7. Wesee that at zero thermal phonon temperature and fixed cavitydrive power P = P = P = 0 . nW, the entanglement de-creases irrespective of the number of thermal photons and per-sists for longer time. Moreover, we see that the critical timeabove which the entanglement disappears remains the samewith varying thermal photons. V. CONCLUSION
We have presented a study of the optical bistability and en-tanglement between two mechanical oscillators coupled to thecavity modes of a two-mode laser via optical radiation pres-sure with realistic parameters. In stark contrast to the usual S-shaped bistability observed in single-mode dispersive optome-chanical coupling, we have found that the optical intensities of n = = = = t E N FIG. 6: Logarithmic negativity E N of the micromechanical mirrorsfor thermal photon numbers N = N = 1 as a function of t atconstant Ω p /γ = 0 . and Ω /γ = 5 for fixed cavity drive laser atP = P = P = 0 . nW with n = n = n = 100 (solid greencurve), n = n = n = 50 (dotdashed magenta line), n = n = n = 10 (blue dotted line) and n = n = n = 5 (red dashed line).See text and the above figures for other parameters. N = = = t E N FIG. 7: Logarithmic negativity E N of the micromechanical mirrorswhen the temperature of the thermal phonon bath is zero, T = 0 K ( n = n = n = 0 ) as a function of t at constant Ω p /γ = 0 . and Ω /γ = 5 for fixed cavity drive laser at P = P = P = 0 . nWwith N = N = N = 100 (solid green line), N = N = N = 50 (dotted red line), and N = N = N = 5 (dotdashed black line) forthe value of the parameter η = − . See text and the above figuresfor other parameters. the two cavity modes exhibit bistabilities for all large valuesof the detuning, due to the parametric amplification-type cou-pling induced by the two-photon coherence. We have also in-vestigated the entanglement of the movable mirror by exploit-ing the intermode correlation induced by the two-photon co-herence. We have here focused on the dynamics of the quan-tum fluctuations of the mirror. We have shown that strongmirror-mirror entanglement can be created in the adiabaticregime. The degree of entanglement E N is significant for a0low thermal noise n and a low cavity-driving laser powers P . The entanglement is supported by direct numerical cal-culations for realistic parameters. Our results suggest that forexperimentally accessible parameters [39, 40], macroscopicentanglement for two movable mirrors can be achieved withcurrent technology and have important implications for quan-tum logic gates based on EIT schemes [54]. Acknowledgments
B. T. gratefully acknowledges numerous discussions withEyob A. Sete. B. T. is supported by the Shanghai Re-search Challenge Fund; New York University Global SeedGrants for Collaborative Research; National Natural ScienceFoundation of China (61571301); the Thousand Talents Pro-gram for Distinguished Young Scholars (D1210036A); andthe NSFC Research Fund for International Young Scientists(11650110425); NYU-ECNU Institute of Physics at NYUShanghai; the Science and Technology Commission of Shang-hai Municipality (17ZR1443600); and the China Science andTechnology Exchange Center (NGA-16-001) and by KhalifaUniversity Internal Research Fund (8431000004).
APPENDIX: COEFFICIENTS IN THE MASTER EQUATION(11)
In this section we derive the coefficients that appear in themaster equation (11) relevant for our system. We follow anidentical procedure to Ref. [25] to obtain the master equation(5) [29, 55, 56]. The next step is to derive the conditioneddensity operators ρ ac = (cid:104) a | ρ AR | c (cid:105) and ρ db = (cid:104) d | ρ AR | b (cid:105) andtheir complex conjugate as appeared in Eq. (5). In the sameway as Ref. [25], we obtain for the matrix elements ddt ρ lk ( t ) = r a ρ (0) lk ρ − i (cid:104) l | [ V , ρ AR ( t )] | k (cid:105) − γ lk ρ lk . (65)Including the spontaneous emission and dephasing process,we can thus determine the equations for ρ ac and ρ db using(65) gives (6 and (7).To study the dynamics of our system, we make use of thelinear approximation by keeping terms only up to the secondorder in the coupling constants, g j ( j = 1 , and consider allorders in the Rabi frequencies in the master equation. The na-ture of the linear approximation means that does not have sat-uration effects in the linear amplification regime. This is jus-tified because the coupling constant of the two quantum fieldsare small as compared to other system parameters occurs onwhich dominates in the time evolution [ ? ]. The zeroth-orderequations of motion for ρ aa , ρ bb , ρ cc , ρ dd , ρ ad , ρ bc , ρ ab , and ρ cd in the coupling constant are ˙ ρ aa ( t ) = r a ρ (0) aa ρ − i Ω( ρ ba − ρ ab ) − γ a ρ aa (66) ˙ ρ bb ( t ) = r a ρ (0) bb ρ − i Ω( ρ ab − ρ ba ) − γ b ρ bb (67) ˙ ρ cc ( t ) = r a ρ (0) cc ρ − i Ω p ( ρ dc − ρ cd ) − γ c ρ cc (68) ˙ ρ dd ( t ) = r a ρ (0) dd ρ − i Ω p ( ρ cd − ρ dc ) − γ d ρ dd (69) ˙ ρ ad ( t ) = r a ρ (0) ad ρ − ( γ ad + i (∆ c − ∆ )) ρ ad − i (Ω ρ bd − Ω p ρ ac ) (70) ˙ ρ bc ( t ) = r a ρ (0) bc ρ − ( γ bc − i (∆ c − ∆ )) ρ bc − i (Ω ρ ac − Ω p ρ bd ) (71) ˙ ρ ab ( t ) = r a ρ (0) ab ρ − ( γ ab + i ∆ c ) ρ ab − i Ω( ρ bb − ρ aa ) (72) ˙ ρ cd ( t ) = r a ρ (0) cd ρ − ( γ cd + i (∆ c − ∆ − ∆ )) ρ cd − i Ω p ( ρ dd − ρ cc ) (73)in which γ j ( j = a, b, c, d ) are the j th atomic-level sponta-neous emission rates and γ ij are the dephasing rates. We nextneed to apply the good-cavity limit where the cavity damp-ing rate is much smaller than the dephasing and spontaneousemission rates. In this limit, the cavity mode variables slowlyvarying than the atomic variables, and thus the atomic vari-ables converge to a steady state quickly. The steady state isfound by setting the time derivatives in (66)-(73) to zero andthe resulting algebraic equations can be solved exactly ρ aa = r a ρd Z aa , ρ bb = r a ρd Z bb ρ ab = r a ρd Z ab ρ cd = r a ρ cd (0) d (cid:48) Z cd ρ cc = r a ρd (cid:48) Z cc ρ dd = r a ρd (cid:48) Z dd Z aa = (cid:0) γ ab + γ b (cid:0) γ ab + ∆ c (cid:1)(cid:1) (1 − η )2 Z ab = i Ω γ b χγ ab + i ∆ c (1 − η )2 Z bb = Ω γ ab (1 − η ) Z cc = (2Ω p γ cd + γ d χ (cid:48) ) (1 + η )2 Z dd = Ω p γ cd (1 + η ) Z cd = i Ω p γ d χ (cid:48) γ cd + i (∆ c − ∆ − ∆ ) (1 + η )2 (74)with d = 2Ω γ ab ( γ a + γ b ) + χγ b γ a , χ = γ ab + ∆ c , χ (cid:48) = γ cd + (∆ c − ∆ − ∆ ) , d (cid:48) = 2Ω p γ cd ( γ c + γ d ) + γ c γ d χ (cid:48) .It proves to be more convenient to introduce a new parameter η ∈ [ − , defined by ρ (0) aa = − η , so that in view of thefact that ρ (0) aa + ρ (0) cc = 1 and the initial coherence takes theform ρ ac (0) = (1 − η ) / . The equations of motion, (6) and(7), can be solved using the adiabatic approximation and theexpressions for ρ ac , ρ cc , ρ bc , ρ ad , ρ bd ρ bb , ρ dd , and ρ ad , so1that (9) are obtained. Here we define: ξ = g AB − AD r a d (cid:48) Z dd ξ = g AB − AD r a d Z bb (75) ξ = g DB − AD r a d (cid:48) Z cc ξ = g DB − AD r a d Z aa (76) ξ = g g BB − AD r a d Z aa ξ = g g BB − AD r a d (cid:48) Z cc (77) ξ = g g BB − AD r a d Z bb ξ = g g BB − AD r a d (cid:48) Z dd (78)where A = − ( γ ac + i ∆ ) − Ω γ bc − i (∆ c − ∆ ) − Ω p γ ad + i (∆ c − ∆ ) (79) B = ΩΩ p γ bc − i (∆ c − ∆ ) + ΩΩ p γ ad + i (∆ c − ∆ ) (80) D = − ( γ bd − i ∆ ) − Ω γ ad + i (∆ c − ∆ ) − Ω p γ bc − i (∆ c − ∆ ) . (81) Substituting (9) into (5) gives the master equation (11) for thecavity modes. [1] S. Bose, K. Jacobs, and P. L. Knight, Phys. Rev. A , 4175(1997).[2] S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Rev. A ,3042 (1997).[3] C. Genes, A. Mari, P. Tombesi, and D. Vitali, Phys. Rev. A ,032316 (2008).[4] D. Vitali, S. Gigan, A. Ferreira, H. R. B¨ohm, P. Tombesi, A.Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, Phys.Rev. Lett. , 030405 (2007).[5] A. Mari and J. Eisert, Phys. Rev. Lett. , 213603 (2009).[6] Jie-Qiao Liao and C. K. Law, Phys. Rev. A , 033820 (2011).[7] C. Genes, D. Vitali, and P. Tombesi,Phys. Rev. A , 050307(R) (2008).[8] H. Ian, Z. R. Gong, Yu-xi Liu, C. P. Sun, and F. Nori, Phys.Rev. A , 013824 (2008); K. Hammerer, M. Aspelmeyer, E. S.Polzik, and P. Zoller, Phys. Rev. Lett. , 020501 (2009).[9] Juan Pablo Paz and Augusto J. Roncaglia, Phys. Rev. Lett. ,220401 (2008).[10] Jie-Qiao Liao, Qin-Qin Wu, and Franco Nori, Phys. Rev. A ,014302 (2014).[11] Michael J. Hartmann and Martin B. Plenio, Phys. Rev. Lett. , 200503 (2008).[12] L. Zhou, Y. Han, J. Jing, and W. Zhang, Phys. Rev. A ,052117 (2011).[13] W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, Phys. Rev. A , 052301 (2013).[14] W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, Phys. Rev. A , 022338 (2013).[15] Stefano Mancini, Vittorio Giovannetti, David Vitali, and PaoloTombesi, Phys. Rev. Lett. , 120401 (2002).[16] Adrian Auer and Guido Burkard, Phys. Rev. B , 235140(2012).[17] M. F. Pereira and I. A. Faragai, Optics express 22 (3), 3439-3446 (2014).[18] M. F. Pereira, Opt Quant Electron 47, 815-820 (2015). [19] M. F. Pereira, Appl. Phys. Lett. 109, 222102 (2016).[20] M. F. Pereira, J. P. Zubelli, D. Winge, A. Wacker A. S. Ro-drigues, V. Anfertev and V. Vaks, Phys. Rev. B , 045306(2017).[21] H. Xiong, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. ,023601 (2005).[22] M. Kiffner, M. S. Zubairy, J. Evers, and C. H. Keitel, Phys. Rev.A , 033816 (2007).[23] S. Qamar, M. Al-Amri, S. Qamar, and M. S. Zubairy, Phys. Rev.A , 033818 (2009).[24] S. Qamar, M. Al-Amri, and M. S. Zubairy, Phys. Rev. A ,013831 (2009).[25] Eyob A. Sete and H. Eleuch, J. Opt. Soc. Am. B , 971-982(2015).[26] A. Tredicucci, Y. Chen, V. Pellegrini, M. Borger, and F. Bassani,Phys. Rev. A , 3493-3498 (1996).[27] A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H.Walther, Phys. Rev. Lett. , 1550 (1983).[28] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod.Phys. , 1391-1452 (2014).[29] M. O. Scully and M. S. Zubairy, Quantum Optics (CambridgeUniversity Press, 1997).[30] Eyob A. Sete, Phys. Rev. A , 063808 (2011).[31] D. F. Walls and G. J. Milburn, Quantum Optics (Springer,2008).[32] E. Alebachew, Opt. Commun. , 133-141 (2007).[33] E. Alebachew, Phys. Rev. A , 023808 (2007).[34] E. A. Sete, Opt. Commun. , 6124-6129 (2008).[35] Wenchao Ge and M Suhail Zubairy, Phys. Scr. , 074015(2015).[36] Claude Cohen-Tannoudji, Jacques Dupont-Roc, Gilbert Gryn-berg, Atom-Photon Interactions: Basic Processes and Applica-tions (Wiley, New York, 2004).[37] J Hald and E S Polzik, J. Opt. B: Quantum Semiclass. Opt. S83 (2001). [38] C. Jiang, H. X. Liu, Y. S. Cui, X. W. Li, G. B. Chen, and X. M.Shuai, Phys. Rev. A , 055801 (2013).[39] S. Simon Gr¨oblacher, Klemens Hammerer, Michael R. Vanner,and Markus Aspelmeyer, Nature , 724-727 (2009).[40] O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heid-mann, Nature , 71-74 (2006).[41] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R.J. Schoelkopf, Rev. Mod. Phys. , 1155 (2010).[42] D. Rist´e, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggel-man, Ya. M. Blanter, K. W. Lehnert, R. N. Schouten, and L.DiCarlo, Nature (London) , 350 (2013).[43] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehn-ert, Science , 710 (2013).[44] Y.-D. Wang and A. A. Clerk, Phys. Rev. Lett. , 253601(2013); L. Tian, ibid . , 233602 (2013); Z.-Q. Yin and Y.-J. Han, Phys. Rev. A , 024301(2009); M. C. Kuzyk, S. J. vanEnk, and H. Wang, ibid . , 062341 (2013); C. Joshi, J. Lar-son, M. Jonson, E. Andersson, and P. ¨Ohberg, ibid . , 033805(2012).[45] A. C. Doherty and K. Jacobs, Phys. Rev. A (4), 2700 (1999).[46] W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC Press, Taylor & Francis Group, LLC, 2016).[47] M. Pinard, A. Dantan, D. Vitali, O. Arcizet, T. Briant, and A.Heidmann, Europhys. Lett. , 747-753 (2005).[48] A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio,Phys. Rev. Lett. , 103605 (2013).[49] G. Vidal and R. F. Werner, Phys. Rev. A (3), 032314 (2002).[50] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A ,022318 (2004).[51] A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian states incontinuous variable quantum information (Bibliopolis, Napoli,2005).[52] Ting Yu,, J. H. Eberly, Science , 598?601 (2009).[53] Qing Lin, Bing He, R. Ghobadi, and Christoph Simon, Phys.Rev. A , 022309 (2014).[54] Amir Feizpour, Greg Dmochowski, and Aephraim M. Stein-berg, Phys. Rev. A , 013834 (2016).[55] W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).[56] F. Kassahun,