Steady-state entanglement and normal-mode splitting in an atom-assisted optomechanical system with intensity-dependent coupling
SSteady-state entanglement and normal-mode splitting in an atom-assistedoptomechanical system with intensity-dependent coupling
Sh. Barzanjeh,
1, 2, ∗ M. H. Naderi, † and M. Soltanolkotabi ‡ Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran School of Science and Technology, Physics Division,Universit`a di Camerino, I-62032 Camerino (MC), Italy Quantum Optics Group, Department of Physics, Faculty of Science,University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran (Dated: August 13, 2018)In this paper, we study theoretically the bipartite and tripartite continuous variable entanglementas well as the normal-mode splitting in a single-atom cavity optomechanical system with intensity-dependent coupling. The system under consideration is formed by a Fabry-Perot cavity with a thinvibrating end mirror and a two-level atom in the Gaussian standing-wave of the cavity mode. Wefirst derive the general form of Hamiltonian describing the tripartite intensity-dependent atom-field-mirror coupling due to the presence of cavity mode structure. We then restrict our treatment to thefirst vibrational sideband of the mechanical resonator and derive a novel form of tripartite atom-field-mirror Hamiltonian. We show that when the optical cavity is intensely driven one can generatebipartite entanglement between any pair of the tripartite system, and that, due to entanglementsharing, the atom-mirror entanglement is efficiently generated at the expense of optical-mechanicaland optical-atom entanglement. We also find that in such a system, when the Lamb-Dicke parameteris large enough one can simultaneously observe the normal mode splitting into three modes.
PACS numbers: 37.30.+i, 03.67.Bg, 42.50.Wk, 85.85.+j
I. INTRODUCTION
Cavity optomechanics is a rapidly growing field of re-search that is concerned with the interaction betweena mechanical resonator (MR) and the radiation pres-sure of an optical cavity field[1–6]. The optomechan-ical coupling widely employed for a large variety ofapplications[7], more commonly as a sensor for the de-tection of weak forces[8] and small displacements[9] oran actuator in integrated electrical, optical, and opto-electronical systems[10, 11]. However, the most exper-imental and theoretical efforts are devoted to coolingand trapping such mechanical resonators to their quan-tum ground state, which more recently have been donesuccessfully[12]. Furthermore, in Ref.[13] the authorshave proposed a different scheme to enhance the coolingprocess by using the photothermal (bolometric) force[14].They have taken into account the noise effects due to thegranular nature of photon absorption and finally haveshown that the mechanical resonator can achieve the low-est phonon occupation number by means of this proce-dure. Moreover, it seems promising for the realization oflong-range interaction between qubits in future quantuminformation hardwares [15], and for probing quantum me-chanics at increasingly large mass and length scales[16].The coupling of a MR via radiation pressure to a cavityfield shows interesting similarities to an intracavity non-linear Kerr-like interaction[6] or even a more complicated ∗ [email protected] † [email protected] ‡ [email protected] form of nonlinearity[17].To observe and control quantum behavior in an op-tomechanical system, it is essential to increase thestrength of the coupling between the mechanical andoptical degree of freedom. However, the form of thiscoupling(e.g., linear or nonlinear) is crucial in determin-ing which phenomena can be observed in such a sys-tem. Thanks to rapid progress of nano-technology, ithas been possible to manipulate the optomechanical cou-pling in quantum optomechanical hybrid systems. In thisdirection most experimental and theoretical efforts aredevoted to entangling a MR either with a single atom[18–22] or with atomic ensembles [23–28], entangling ananomechanical oscillator with a Cooper-pair box [29],and entangling two charge qubits [30] or two Josephsonjunctions [31] via nanomechanical resonators. Alterna-tively, schemes for entangling a superconducting coplanarwaveguide field with a nanomechanical resonator, eithervia a Cooper-pair box within the waveguide [32], or viadirect capacitive coupling [33], have been proposed.In Ref.[24] the authors have proposed a scheme for therealization of a hybrid, strongly quantum-correlated sys-tem consisting of an atomic ensemble surrounded by ahigh-finesse optical cavity with a vibrating mirror. Theyhave shown that, in an experimentally accessible pa-rameter regime, the steady state of the system showsboth tripartite and bipartite continuous variable(CV) en-tanglement. More recently, the dynamics of a movablemirror of a cavity coupled through radiation pressureto the light scattered from ultracold atoms in an opti-cal lattice has been investigated Ref.[34]. The authorhas shown that in the presence of atom-atom interactionas a source of nonlinearity[35], the coupling of the me- a r X i v : . [ qu a n t - ph ] D ec chanical oscillator, the cavity field fluctuations and thecondensate fluctuations (Bogoliubov mode) leads to thesplitting of the normal mode into three modes (normal-mode splitting(NMS)[36–40]). The system describedthere shows a complex interplay between distinctly threesystems namely, the nanomechanical cantilever, opticalmicrocavity and the gas of ultracold atoms.The optomechanical NMS is one of the fascinating phe-nomena arising from the strong coupling between the cav-ity and the mechanical mirror[41–43]. In Ref.[42] it hasbeen shown that the cooling of mechanical oscillators inthe resolved sideband regime at high driving power lasercan entail the appearance of NMS. Moreover, the dy-namics of a movable mirror of a nonlinear optical cavityis considered in Ref.[43]. It has been shown that a χ (3) medium with a strong Kerr nonlinearity placed insidethe cavity inhibits the NMS due to the photon blockademechanism(this just happens only if the Kerr nonlinear-ity is much greater than the cavity decay rate). As theauthors have shown in Refs.[34] and [43] the nonlinear-ity plays a crucial role in the appearance of NMS in theoptomechnical systems.The main purpose of the present paper is to studythe quantum behavior of an atom-assisted cavity op-tomechanical system in which a single two-level atom istrapped in the standing-wave light field of a single-portFabry-Perot cavity. The infinite set of optical modes ofthe cavity can be described by Hermit-Gauss modes. Aswe will see, the intracavity mode structure can be em-ployed to realize a type of intensity-dependent couplingof the single atom to the vibrational mode of MR. Thepresence of such intensity-dependent interaction modifiesthe dynamics of the system, the entanglement propertiesand the displacement spectrum of MR. We show that inthe first vibrational sideband of MR, the stationary, i.e.,long-lived, atom-mirror entanglement can be generatedby proper matching the Lamb-Dicke parameter(LDP).This parameter plays an important role in our investi-gation in the sense that it determines the strength ofthe nonlinearity in the system. We show that the bi-partite entanglement between the subsystems extremelydepends on the LDP. It is also remarkable that, in thesteady-state condition, the high resolution of NMS in theform of three-mode splitting is approached. In particu-lar, the appearance of intensity-dependent coupling leadsto a progressive increasing of NMS due to the strong non-linear atom-field-mirror interaction.The paper is organized as follows. In Sec. II we derivean intensity-dependent Hamiltonian describing the triplecoupling of atom-field-mirror through j -phonon excita-tions of the vibrational sideband. In Sec. III, we derivethe quantum Langevin equations (QLEs) and linearizethem around the semiclassical steady state. In Sec. IVwe study the steady state of the system and quantifythe entanglement properties of the system by using thelogarithmic negativity. In Sec. V we investigate the ap-pearance of NMS in the displacement spectrum of themirror. Our conclusions are summarised in Sec. VI. II. MODEL
The system studied in this paper is sketched in Fig. 1.It consists of a hybrid system formed by a single two-level atom with transition frequency ω e which trapped inthe standing-wave light field of a single-port Fabry-Perotcavity with a movable mirror coated on the plane side ofa mechanical resonator. The geometry of the resonatordetermines the spatial structure of the acoustic modes.The movable mirror is treated as a quantum mechani-cal harmonic oscillator with effective mass m , frequency ω m , and energy decay rate γ m . The system is also coher-ently driven by a laser field with frequency ω l throughthe cavity mirror with amplitude E . We assume that thesingle atom is indirectly coupled to the mechanical os-cillator via the common interaction with the intracavityfield with frequency ω c . FIG. 1. The schematic of the atom-assisted optomechanicalsystem. It contains an optical cavity ended with a fixed mirrorand a slightly moving mirror which is attached to a spring.Inside the cavity there is a two-level atom. The system iscoherently driven by a laser field
In our investigation we can restrict the model to thecase of single-cavity and mechanical modes. This is jus-tified when the cavity free spectral range is much largerthan the mechanical frequency ω m (i.e., not too large cav-ities). In this case, scattering of photons from the drivenmode into other cavity modes is negligible [44] and the in-put laser successfully drives only one cavity mode. Thisguarantees the fact that only one cavity mode partici-pates in the optomechanical interaction and the neigh-bouring modes are not excited by a single central fre-quency input laser. In addition, one can restrict to asingle mechanical mode when the detection bandwidth ischosen such that it includes only a single, isolated, me-chanical resonance and the mode-mode coupling is neg-ligible [45]. A. Hamiltonian of the system
In the absence of dissipation and fluctuations, the totalHamiltonian of the system is given by the sum of threeterm; the free evolution term[19, 24] H = (cid:126) ω c a † a + (cid:126) ω m b † b + (cid:126) ω e σ z , (1)the interaction term H int = − (cid:126) ξ ( b + b † ) a † a + (cid:126) χ mnl ( (cid:126)r , x )[ aσ + + h.c (cid:3) , (2)and the laser driven term H dri = i (cid:126) E ( a † e − i ω l t − ae i ω l t ) , (3)where a ([ a, a † ] = 1) is the annihilation operator of thecavity field with the decay rate κ , b ([ b, b † ] = 1) is themotional annihilation operator of the MR, and the singletwo-level atom is described by the spin-1 / σ − , σ + and σ z which satisfy the commu-tation relations[ σ + , σ − ] = σ z and [ σ z , σ ± ] = ± σ ± . Itshould be noted that the free Hamiltonian(1) has beenwritten within the Raman-Nath approximation[51], i.e.,in the limit when the atom is allowed only to move overa distance which is much less than wavelength of thelight. Therefore, in this approximation, one can ne-glect the kinetic energy of the atom. The first term of H int is the optomechanical coupling with the radiation-pressure coupling constant ξ = ( ω c /L ) x ZPF , in which x ZPF = (cid:112) (cid:126) /mω m is the zero point fluctuations of me-chanical oscillator. The second term of H int denotes the”three-body” interactions among the atom, the cavityfield and the vibration of the mirror. The field-atomcoupling rate in terms of an infinite set of optical modesis well described by the Hermite-Gauss modes[46, 47] χ mnl ( (cid:126)r ) = g K mnl ( x, y, z )sin (cid:104) ψ mnl ( x, y, z ) − lπ (cid:105) , (4)where, for m, n = 0 , ... , l = 1 , , ... , K mnl ( x, y, z ) = H n [ √ yw ( x ) ] H m [ √ zw ( x ) ]exp[ − z + y w ( x ) ] w ( x ) √ π n + m − m ! n ! L , (5) ψ mnl ( x, y, z ) = kx − φ ( x )( m + n + 1) + k z + y R ( x ) , (6)Here g = µ (cid:112) ω c /(cid:15) V , (cid:15) is the vacuum permittivity, V shows the volume of the cavity and µ is the electric-dipoletransition matrix element. H n ( y ) is the n -th Hermitepolynomial, w ( x ) = w (cid:104) xx R ) (cid:105) is the beam waistat x which is defined as the distance out from the axiscenter of the beam where the irradiance drops to 1 /e of its values on axis, R ( x ) = x + x R /x is the radius ofcurvature of the wavefront at x , φ ( x ) = arctan( x/x R )is the Gouy phase shift[46], w is the cavity waist ra-dius which depends on the geometry of the Fabry-Perotcavity and x R = w k/ √ w . Thecoupling rate χ mnl ( (cid:126)r , x ) depends on the initial atomicposition (cid:126)r (measured from the cavity waist) as well asthe displacement x = x ZPF ( b + b † ) of the mirror due to k = ω eff ( x ) /c , where ω eff ( x ) = ω c (1 − xL ). As we willsee in the next section, this dependence to the positionof MR is responsible for the appearance of a new typeof optomechanical nonlinearity. Finally, the Hamiltonian H dri describes the input driving by a laser with frequency ω l and amplitude |E| = (cid:112) κP/ (cid:126) ω l , where P is the inputlaser power and κ is the cavity loss rate through its inputport. B. The nonlinear atom-field-mirror coupling
As we have seen the Gaussian standing-wave struc-ture of the cavity mode leads to the field-atom couplingrate χ mnl ( r , x ). Such field-atom coupling in the pres-ence of the mode structure of the field has been stud-ied extensively in the literature and it has been shownthat a certain type of nonlinearity is prepared in thefield-atom system. For instance, in Refs.[48, 49] the in-fluences of the atomic motion and the field-mode struc-ture on the atomic dynamics have been investigated. Ithas been shown that the atomic motion and the field-mode structure give rise to nonlinear transient effects inthe atomic population which are similar to self-inducedtransparency and adiabatic effects. In our treatment,the spatial field-mode structure leads to the appearanceof an intensity-dependent interaction among the intra-cavity optical mode, the MR and the single atom. Toshow this, we assume that the atom is well located at thetransverse (polar) coordinate(measured from the cylin-drically symmetric cavity axis along the x direction) ρ = (cid:112) x + y = µ w ( x ) where 0 ≤ µ ≤
1. In the x direction the localization of the atom can be expressedas k x = (cid:15)π for (cid:15) >
0, where k = ω c /c .At lowest order of the optical modes, i.e., m = n =0 , l = 1, the tripartite coupling rate reduces to χ ( (cid:126)r , x ) ≡ χ ( x ) = 2 g e µ w ( x ) √ πL sin (cid:104) kx − φ ( x ) − π µx kw (cid:105) , (7)which can be rewritten in terms of the mirror positionby using the position dependence of the wavelength k = k (1 − x/L ) as χ ( x ) = 2 g e µ w ( x ) √ πL sin (cid:104) θ + η x (cid:105) , (8)where η = µx w k L and θ = (1 + 2 µw k ) k x − φ ( x ) − π . (9)By substituting x = x ZPF ( b + b † ) in Eq.(8) we obtain χ ( b, b † ) = g ie µ w ( x ) √ πL (cid:110) e iθ exp[ iη ( b + b † )] − h.c (cid:111) , (10)where the parameter η = η x ZPF = 2 πµ(cid:15)w k L (cid:114) (cid:126) mω m , (11)is the so-called ”Lamb-Dicke parameter”(LDP). By usingthe Baker-Campbell-Hausdorff theorem in Eq.(10) andexpanding the exponential terms in terms of b and b † ,the coupling rate can be written as χ ( b, b † ) = g e − η / ie µ w ( x ) √ πL (cid:110) e iθ (cid:88) m,m (cid:48) ( iηb † ) m (cid:48) ( iηb ) m m ! m (cid:48) ! − h.c (cid:111) . (12)By using the bosonic commutation relation of the oper-ators b and b † , the j -th term of the field-atom couplingrate is obtained as follows χ j ( b, b † )= g e − η / ie µ w ( x ) √ πL (cid:104) e iθ (cid:88) m ( iη ) m + j ( b † ) m + j b m m !( m + j )! − h.c (cid:105) = g j,µ ( b † ) j f j ( n b ) + h.c, (13)where g j,µ = g e − η / ( iη ) j ie µ w ( x ) √ πL e iθ describes the effective atom-field-mirror coupling rate and the Hermitian nonlinearityfunction f j ( n b ) defined by f j ( n b ) = (cid:88) m ( iη ) m n b ! m !( m + j )!( n b − m )! = n b !( n b + j )! L jn b ( − η ) , (14)with n b = b † b and L jn b ( − η ) as the associated Laguerrepolynomial, describes a nonlinear atom-field-mirror cou-pling through j -phonon excitations of the vibrationalsideband. The nonlinearity function f j ( n b ) has a cen-tral role in our treatment. It determines the form ofnonlinearity of the intensity dependence of the couplingamong the cavity field, the MR and the single atom. Aswe will see, this function drastically influences the dy-namics of the system, its entanglement properties andits responsible for the appearance of NMS with high vis-ibility in the displacement spectrum of the MR. Fig.2(a)shows the nonlinearity function f j ( n b ) as a function of n b and for different values of phonon excitation number j . As is seen, this function has maximum contributionaround small values of vibrational acoustic excitation n b .Furthermore, by increasing the number j the strength ofthe nonlinearity function f j ( n b ) decreases considerably.On the other hand, Fig.2(b) shows that the nonlinearitydecreases by increasing the LDP. It is remarkable that,for the higher orders of the vibrational sideband, j ≥ H ( j )int = − (cid:126) ξ ( b + b † ) a † a + (cid:126) (cid:104) g j,µ ( b † ) j f j ( n b )+ h.c (cid:105)(cid:104) aσ + + h.c (cid:105) . (15) j=1 j=2 j=3 j=4 (a) n b f j (cid:72) n b (cid:76) j=1 j=2 j=3 j=4 (b) Η f j (cid:72) n b (cid:76) FIG. 2. The nonlinearity function f j ( n b ) as a function of:(a)phonon number n b for η = 0 .
08 and for different values ofvibrational sideband j and (b) the Lamb-Dicke parameter, η ,for n b = 10 and for different values for vibrational sideband j . Near the photon-phonon resonance [19] where the fre-quencies satisfy ω m + ω e − ω c (cid:39)
0, the rotating-waveapproximation reduces the above Hamiltonian to H ( j )int (cid:39) − (cid:126) ξ ( b + b † ) a † a + (cid:126) (cid:104) g j,µ ( b † ) j f j ( n b ) aσ + + h.c (cid:105) . (16)This Hamiltonian describes a nonlinear tripartite atom-field-mirror coupling and represents a novel type ofoptomechanical intensity-dependent interaction. TheHamiltonian (16) is general and one can recover the re-sults of Ref.[24] by taking j = 0, f j ( n b ) → j = 1, f j ( n b ) →
1. Tostudy the system dynamics we restrict our investigationby considering the first excitation of the vibrational side-band i.e., j = 1. In this limit one may use the followingsimple form of the Hamiltonian H ( j =1)imt (cid:39) − (cid:126) ξ ( b + b † ) a † a + (cid:126) g µ (cid:104) b † f ( n b ) aσ + + σ − a † f ( n b ) b (cid:105) , (17)where f ( n b ) ≡ f ( n b ) and g µ = g e − η / ηe µ w ( x ) √ πL . Since wedeal with a well localized atom we can assume θ = π inEq.(9), which is realized by choosing a proper value of (cid:15) corresponding to the position of atom in the x direc-tion. We pointed out that for the experimentally feasibleparameters of the system under consideration[50], i.e., k (cid:39) m − , m = 10pg, ω m / π = 10MHz, L = 1 µ mand for the smallest achievable value of the cavity beamwaist w ≥ − m the LDP is always less than one, i.e., η <
1. In this limit we can keep terms up to first or-der in the phonon number n b , and safely truncate thesummation of Eq.(14), f j =1 ( n b ) (cid:39) − η n b . (18)By substituting Eq.(18) into the Hamiltonian (17) onecan write the interaction of Hamiltonian as H int = − (cid:126) ξ ( b + b † ) a † a + (cid:126) g µ (cid:104) b † aσ + + σ − a † b (cid:105) −− (cid:126) η g µ (cid:104) b † n b aσ + + σ − a † n b b (cid:105) . (19)Note that the first and second terms of the above Hamil-tonian denote the standard tripartite atom-field-mirrorcoupling which recently has been studied in Refs.[19] and[21]. The third term denotes an intensity-dependent cou-pling among the three subsystems of atom-field-mirror.This type of nonlinear coupling is attributed to the spa-tial field-mode structure at the position of the atom. III. DYNAMICS OF THE SYSTEM
To describe the dynamical behavior of the systemunder consideration it is necessary to consider thefluctuation-dissipation processes affecting the three sub-systems. For this purpose, we first assume the exci-tation probability of the single atom to be small. Inthis limit, the dynamics of the atomic polarization canbe described in terms of the bosonic operators c and c † ([ c, c † ] = 1) [24, 52], where the atomic annihilation op-erator is defined as c = σ − / (cid:112) |(cid:104) σ z (cid:105)| . This is valid in thelow atomic excitation limit, i.e., when the atom is ini-tially prepared in its ground state[24]. This means thatthe single-atom excitation probability should be muchsmaller than one. i.e., g | α s | ∆ a + γ a (cid:28)
1, where ∆ a = ω a − ω l is the atomic detuning with respect to the laser and γ a is the decay rate of the excited atomic level. Therefore,the bosonization of the atomic operators is valid only if g (cid:28) ∆ a + γ a that is the atom is weakly coupled to thecavity.The dynamics of the system is fully characterized bythe following set of nonlinear quantum Langevin equa-tions, written in the frame rotating at the input laserfrequency,˙ c = − [ γ a + i∆ a ] c − i G (1 − η n b ) ab † + (cid:112) γ a F a , (20a)˙ a = − [ κ + i∆ f ] a + i ξ a ( b + b † ) − i G (1 − η n b ) bc ++ E + √ κa in , (20b)˙ b = − [ γ m + i ω m ] b + i ξ a † a − i G [(1 − η n b ) ac † − η a † cb ] ++ (cid:112) γ m b in , (20c) where ∆ f = ω c − ω l is the cavity detuning with respectto the laser, G = g µ (cid:112) |(cid:104) σ z (cid:105)| and γ m is the decay rate ofthe vibrational mode of the MR. The motional quantumfluctuation b in ( t ) satisfies the following relations[53] (cid:104) b in ( t ) b † in ( t (cid:48) ) (cid:105) = [ (cid:104) n b,th (cid:105) + 1] δ ( t − t (cid:48) ) , (cid:104) b † in ( t ) b in ( t (cid:48) ) (cid:105) = (cid:104) n b,th (cid:105) δ ( t − t (cid:48) ) , (21) (cid:104) b in ( t ) b in ( t (cid:48) ) (cid:105) = (cid:104) b † in ( t ) b † in ( t (cid:48) ) (cid:105) = 0 , where (cid:104) n b,th (cid:105) is the mean number of phonons in theabsence of optomechanical coupling, determined by thetemperature of the mechanical bath T , (cid:104) n b,th (cid:105) = 1 e (cid:126) ωmkBT − . (22)The only nonvanishing correlation function of the noisesaffecting the atom and the cavity field is (cid:104) a in ( t ) a † in ( t (cid:48) ) (cid:105) = (cid:104) F a ( t ) F † a ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) )[53]. A. Linearization of QLEs
Our aim is to study the conditions under which one canefficiently correlate and entangle the atom and the me-chanical resonator by means of the common interactionwith the intracavity optical mode. As shown in Refs.[45]and [54], a straightforward way for achieving stationaryand robust entanglement in continuous variable optome-chanical systems, is to choose an operating point wherethe cavity is intensely driven so that the intracavity fieldis strong, which is realized for high-finesse cavities andenough driving power. Therefore, we focus onto the dy-namics of the fluctuations around the classical steadystate by decomposing each operator in Eqs. (20) as thesum of its steady-state value and a small fluctuation, e.g., a = α s + δa , b = β s + δb , and c = c s + δc . The steadystate terms of these operators are given by b s = α s ( ξ/ − G )( ω m − i γ m ) , (23a) c s = G α s (i γ a − ∆ a ) , (23b) E = α s (cid:104) i∆ f + κ − | G | ( γ a + i∆ a ) (cid:105) , (23c)where ∆ f = ∆ f − ξ Re ( b s ) denotes the effective op-tomechanical detuning and ξ = 2 ξ a s . The other param-eters are defined in the appendix. In the linearizationmanner, we also obtain the following linear QLEs for thequantum fluctuations of the triple system δ ˙ c = − [ γ a + i∆ a ] δc − i G (cid:104) (1 − η | b s | )( a s δb † + b ∗ s δa ) − η a s b ∗ s ( b s δb † + b ∗ s δb ) (cid:105) + (cid:112) γ a F a , (24a) δ ˙ a = − [ κ + i∆ f ] δa + i ξ (cid:104) δa ( b s + b ∗ s ) + a s ( δb + δb † ) (cid:105) − i G (cid:104) (1 − η | b | )( c s δb + b s δc ) − η b s c s ( b s δb † + b ∗ s δb ) (cid:105) + √ κa in , (24b) δ ˙ b = − [ γ m + i ω m ] δb + i ξ a s ( δa † + δa ) − i G (cid:104) (1 − η | b s | )( a s δc † + c ∗ s δa ) − η a s c ∗ s ( b s δb † + b ∗ s δb ) − η { b s ( c s δa † + a s δc ) + 2 a s b s c s δb } (cid:105) + (cid:112) γ m b in , (24c)in terms of the fluctuations of the quadrature operators, δX a = 1 √ δa + δa † ) , δY a = 1 √
2i ( δa − δa † ) , (25) δX c = 1 √ δc + δc † ) , δY c = 1 √
2i ( δc − δc † ) , (26) δq = 1 √ δb + δb † ) , δp = 1 √
2i ( δb − δb † ) . (27)The resulting linearized QLEs can be written in the fol-lowing compact matrix form˙ u ( t ) = Au ( t ) + n ( t ) , (28)where u ( t ) = (cid:2) δq ( t ) , δp ( t ) , δX a ( t ) , δY a ( t ) , δX c ( t ) , δY c ( t ) (cid:3) T is the vector of CV fluctuation operators and n ( t ) = [ √ γ m q in ( t ) , √ γ m p in ( t ) , √ κX in a ( t ) , √ κY in a ( t ) , √ γ a X in c ( t ) , √ γ a Y in c ( t )] T is the corresponding vector ofnoises. Moreover, the drift matrix A is a 6 × A = − Γ Ω − M I M R − M I M R − Ω − Γ − M R − M I − M R − M I − G I G R − κ ∆ f − G I G R ξ − G R − G I − ∆ f − κ − G R − G I − N I N R − N I N R − γ a ∆ a − N R − N I − N R − N I − ∆ a − γ a , (29)where O Ri and O Ii denote the real and imaginary parts ofparameter O i , respectively. The other matrix elementsare defined in the appendix. B. Stationary quantum fluctuations
Here, we focus our attention on the stationary proper-ties of the system. For this purpose we should considerthe steady state condition governed by Eq. (28). Thesteady state is reached when the system is stable, whichoccurs if and only if all the eigenvalues of the matrix A have a negative real part. These stability conditions canbe obtained, for example, by using the Routh-Hurwitzcriterion[55]. The steady state is a zero-mean Gaussian state due tothe fact that the dynamics of the fluctuations is linearizedand all noises are Gaussian. As a consequence, it is fullycharacterized by the 6 × V ij = (cid:104) u i ( ∞ ) u j ( ∞ ) + u j ( ∞ ) u i ( ∞ ) (cid:105) . (30)The formal solution of Eq.(28) yields[45] V ij = (cid:90) ∞ ds (cid:90) ∞ ds (cid:48) M ik ( s ) M jl ( s (cid:48) ) D kl ( s − s (cid:48) ) , (31)where M ( t ) = exp( At ) and D ( s − s (cid:48) ) is the diffu-sion matrix, the matrix of noise correlations, defined as D kl ( s − s (cid:48) ) = (cid:104) n k ( s ) n l ( s (cid:48) ) + n l ( s (cid:48) ) n k ( s ) (cid:105) /
2. For the noisediffusion matrix we have D ( s − s (cid:48) ) = Dδ ( s − s (cid:48) ), where D = diag[ γ m (2¯ n b + 1) , γ m (2¯ n b + 1) , κ, κ, γ a , γ a ]. There-fore, Eq.(31) is simplified to V = (cid:90) ∞ dωV ( ω ) , (32)where V ( ω ) = M ( ω ) DM ( ω ) T . (33)When the stability conditions are satisfied ( M ( ∞ ) = 0),one can obtain the following Lyapunov equation AV + V A T = − D. (34)Equation (34) is a linear equation for V and can bestraightforwardly solved. However, the explicit form of V is complicate and will not be reported here. IV. ENTANGLEMENT PROPERTIES OF THESTEADY-STATE OF THE TRIPARTITE SYSTEM
In this section we examine the entanglement propertiesof the steady state of the tripartite system under consid-eration. For this purpose, we consider the entanglementof the three possible bipartite subsystems that can be ob-tained by tracing over the remaining degrees of freedom.Such bipartite entanglement will be quantified by usingthe logarithmic negativity [56], E N = max[0 , − ln2 η − ] , (35)where η − ≡ − / (cid:104) Σ( V bp ) − (cid:112) Σ( V bp ) − V bp (cid:105) / isthe lowest symplectic eigenvalue of the partial transposeof the 4 × V bp , associated with the selected bipar-tition, obtained by neglecting the rows and columns ofthe uninteresting mode, V bp = (cid:18) B CC T B (cid:48) (cid:19) , (36) E Nfa E Nam E Nmf (a) (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) a (cid:144) Ω m E N (b) E Nfa E Nam E Nmf (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) a (cid:144) Ω m E N FIG. 3. (Color online) Plot of E N of the three bipartite sub-systems [ E amN (atom-mirror), E faN (field-atom), E mfN (mirror-field)]versus the normalized atomic detuning ∆ a /ω m at fixedtemperature T = 0 . η = 0 .
04 and (b) η = 0 .
08 . The optical cavitydetuning has been fixed at ∆ f = − ω m , while the other pa-rameters are ω m / π = 10 MHz, Q = 11 × , m = 10 pg, κ = 0 . ω m , k = 10 m − , P c = 800 µW , γ a / π = 0 . ω m ,and g / π = 10 Hz. and Σ( V bp ) ≡ det B + det B (cid:48) − C .In Fig.3 we have plotted the three bipartite logarith-mic negativities, E amN (atom-mirror), E faN (field-atom),and E mfN (mirror-field) versus the normalized atomic de-tuning ∆ a /ω m at fixed temperature T = 0 . η = 0 .
04, Fig.3aand η = 0 .
08, Fig.3b] and for the experimentally fea-sible parameters[50], i.e., a mechanical resonator withoscillation frequency ω m / π = 10 MHz, quality factor Q = 11 × , m = 10 pg and an optical cavity withlength L = 1 µ m and damping rate κ = 0 . ω m drivenby a laser with k (cid:39) m − and power P c = 800 µW. Theatom damping constant has been taken γ a / π = 0 . ω m with coupling constant g / π = 10 Hz.The optical cavity detuning has been fixed at ∆ f = − ω m which turns out be the most convenient choice.As seen, by increasing LDP, the two bipartite entangle-ment of E amN and E faN increase and the bipartite entan-glement of E mfN decreases overall. The reason is that,by increasing the LPD the tripartite atom-field-mirrorcoupling rate increases compared to the coupling rate of the bipartite field-mirror subsystem, or equivalently theparameter G/ξ is increased. This result reveals that bychanging the LDP one can control the tripartite couplingamplitude or even goes through the regime in which thetripartite system reduces to an effective bipartite subsys-tem. However, the three logarithmic negativities do notbehave in the same way and the entanglement sharing isevident. In particular, the entanglement of interest, i.e., E amN , increases at the expense of the mirror-field entan-glement, while E faN remains always non-negligible. (a) E Nfa E Nam E Nmf (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) a (cid:144) Ω m E N (b) E Nfa E Nam E Nmf (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) a (cid:144) Ω m E N FIG. 4. (Color online) Plot of E N of the three bipar-tite subsystems [ E amN (atom-mirror), E faN (field-atom), E mfN (mirror-field)]versus the normalized atomic detuning ∆ a /ω m for a fixed value of the LDP, η = 0 .
04, and for two differenttemperatures: (a) T = 1 .
2K and (b) T = 3K. The opticalcavity detuning has been fixed at ∆ f = − ω m and the otherparameters are as in Fig. 3. Fig. 4 shows the logarithmic-negativity of the three bi-partite subsystems versus the normalized atomic detun-ing ∆ a /ω m for a fixed value of the LDP, η = 0 .
04, andfor two different temperatures: T = 1 . T = 3 K (Fig.4b). The optical cavity detuning has beenagain fixed at ∆ f = − ω m . As expected, the three kindsof bipartite entanglement decrease by increasing temper-ature, but the atom-mirror and field-atom entanglementshow highly temperature robustness. However, the field-mirror entanglement shows extremely fragile entangle-ment robustness versus temperature and its logarithmic-negativity falls down to zero at T = 3K.Generally, the scheme is able to generate apprecia-ble entanglement between the atom and the MR, es-pecially by sharing from the mirror-field entanglement.Similar bipartite entanglement behavior can be observedin other similar tripartite systems, such as the atom-field-mirror scheme proposed in Ref. [24], the microwave-optical-mirror system of Ref. [59] and the two cavity op-tomechanical setup of Ref. [60]. E Nam E Nfa E Nmf Η E N FIG. 5. (Color online) Plot of E N of the three bipartite sub-systems [ E amN (atom-mirror), E faN (field-atom), E mfN (mirror-field)]versus the LDP, η , for fixed temperature T = 0 . f = − ω m and∆ a = ω m , respectively. The other parameters are as in Fig.3. The effect of the LDP is also illustrated in Fig. 5, wherewe have plotted the logarithmic-negativity as a functionof η at fixed optical cavity detuning ∆ f = − ω m and atatomic detuning ∆ a = ω m . It is clear that by increasingthe LDP, the field-atom entanglement increases when theentanglement between the intracavity mode and the MRis drastically suppressed. We also observe that the atom-mirror logarithmic-negativity E amN plateaus in a case as η increases.Fig.6 shows E faN and E amN versus ∆ a /ω m and γ a /ω m for η = 0 .
04. As is clear, the E faN and E amN are max-imized around sideband ∆ a (cid:39) ω m . By increasing theatomic spontaneous emission rate γ a as we expected,both logarithmic-negativities decrease drastically.The entanglement features of the tripartite system atthe steady state can be observed by experimentally mea-suring the corresponding CM. This can be done by com-bining existing experimental techniques. By homodyn-ing the cavity output one can measure the cavity fieldquadratures. Ref.[3] has proposed a scheme to measuremechanical position and momentum of the MR, in whichby adjusting the detuning and bandwidth of an addi-tional adjacent cavity, both position and momentum ofthe mirror can be measured by homodyning the outputof the second cavity. Moreover, by adopting the samescheme of Ref.[61], the atomic polarization quadratures X a and Y a can be measured, i.e., by making a Stokes pa-rameter measurement of a laser beam, shined transversalto the cavity and to the cell and off-resonantly tunedto another atomic transition. Very recently, Ref.[62] hasdemonstrated the proof of principle of the use of a Bose- FIG. 6. (Color online) Density plot of E N of the bipartitesubsystems: (a) E faN and (b) E amN versus ∆ a /ω m and γ a /ω m for η = 0 .
04 and for T = 0 . f = − ω m . The other parameters are as inFig. 3. Einstein condensate(BEC) as a diagnostic tool to deter-mine the elusive mirror-light entanglement in a hybridoptomechanical device. In such a case, one dose find aworking point such that the mirror-light entanglement isreproduced by the BEC- light quantum correlations.
V. NORMAL-MODE SPLITTING IN THEDISPLACEMENT SPECTRUM OF THE MR
In this section, we show that the atom-field-mirror cou-pling leads to the splitting of the normal mode into threemodes [Normal Mode Splitting(NMS)]. The optomechan-ical NMS however involves driving four parametricallycoupled nondegenerate modes out of equilibrium. TheNMS does not appear in the steady state spectra butrather manifests itself in the fluctuation spectra of themirror displacement. To study the NMS in our systemwe need to find out the displacement spectrum of mirroras: S q ( ω ) = 12 π (cid:90) d Ω e − i ( ω +Ω) t (cid:104) δq ( ω ) δq (Ω) + δq (Ω) δq ( ω ) (cid:105) = V ( ω ) , (37)where V ( ω, Ω) = 1 / (cid:104) δq ( ω ) δq (Ω) + δq (Ω) δq ( ω ) (cid:105) is anelement of CM which is given by Eq.(33). Unfortunately,the analytical form of the displacement spectrum of themirror is too complicate to put a clear physical interpre-tation on it. Thus, in the following, we give and analyzethe results obtained by numerical calculations. Η =0.016 Η =0.04 (cid:45) (cid:45) Ω (cid:144) Ω m S q FIG. 7. (Color online) Normalized plot of the displacementspectrum S q ( ω ) versus ω/ω m at fixed temperatures T = 0 . η = 0 . η = 0 . f = ω m and ∆ a = ω m , respectively. Theother parameters are as in Fig. 3. Fig. 7 shows the displacement spectrum of the MR asa function of the normalized frequency ω/ω m at ∆ f = ω m , ∆ a = ω m and for two different values of the LDP: η = 0 . η = 0 .
04. For the small values of the LDP, weobserve the usual normal-mode splitting into two modeswith central peaks at the sidebands ω = ± ω m . Thisfigure shows a highly symmetric structure with respectto ω = 0. As is seen, by increasing the LDP the normalmode splits up into three modes.A more clear illustration of the three-mode splittingis shown in Fig.8. This figure shows the displacementspectrum of the MR versus the normalized frequency ω/ω m and atomic detuning ∆ a /ω m at ∆ f = ω m . Thethree-mode splitting manifests itself mainly at ∆ a (cid:39) ω .By going through the region far from ∆ a (cid:39) ω , three-mode splitting merges into two-mode splitting around∆ a (cid:39) . ω m and ∆ a (cid:39) . ω m . The NMS is associatedwith the mixing among the vibrational mode of the MR, FIG. 8. (Color online) Density plot of the displacement spec-trum S q ( ω ) versus ω/ω m and ∆ a /ω m for T = 0 .
4K and η = 0 .
04. The optical cavity detuning has been fixed at∆ f = ω m . The other parameters are as in Fig. 3. the fluctuations of the cavity field around the steady stateand the fluctuations of the atomic mode. The origin ofthe fluctuations of the cavity field is the beat of the pumpphotons with the photons scattered from the atom. Fornot so large values of the LDP(small nonlinearity) thefield-atom coupling is much smaller than the field-mirrorcoupling. Therefore, the system simply reduces to thecase of two mode coupling, i.e., coupling between themechanical mode and the photon fluctuations[34]. Whenthe LDP is large enough the mechanical mode, the pho-ton mode, and the atomic mode forms a system of threecoupled oscillators. The occurrence of splitting of thenormal mode into three modes has been analyzed re-cently in another tripartite system, i.e., a cavity quantumoptomechanical system of ultracold atoms in an opticallattice[34]. Furthermore, similar three coupled oscilla-tor experimental results where two coupled cavities, eachcontaining three identical quantum wells[63] and one mi-crocavity containing two quantum wells[64] have been re-ported.It should be pointed out that to observe the NMS, theenergy exchange between the three modes should takeplace on a time scale faster than the decoherence of eachmode. The normal mode splitting into three modes dueto local increasing of the LDP has also been reported inRef.[65], where the authors have shown that the NMScan be observed only if the coupling between the atomsand the cavity is strong enough. This strong couplingcan be achieved by increasing the atom numbers. Oneexperimental limitation could be spontaneous emissionwhich leads to momentum diffusion and hence heating ofthe atomic sample[66]. In our model, we don’t encountersuch a limitation and three-mode splitting is approachedby proper choosing of the LDP.0 VI. CONCLUSION
In this work, we have proposed a theoretical schemefor the realization of tripartite intensity-dependent cou-pling among a single mode of a Fabry-Perot cavity withan oscillating mirror, a single two-level atom inside it,and a vibrational mode of the oscillating mirror. Wehave shown that in the presence of Gaussian standing-wave of the optical cavity mode, a type of tripartite be-tween atom-mirror-field coupling can be manifested. Todescribe such interaction we then have found the generalform of the corresponding nonlinear Hamiltonian. Wehave restricted our investigation to first vibrational side-band j = 1 and have studied its dynamics by adoptinga QLE treatment. We have focused our attention onthe steady state of the system and in particular, on thestationary quantum fluctuations of the system by solv-ing the linearized dynamics around the classical steadystate. We have seen that, in an experimentally acces-sible parameter regime, the steady state of the systemshows both the tripartite and the bipartite CV entan-glement. We have shown that the LDP(as a measure ofthe strength of nonlinearity)can extremely modifies boththe tripartite and the bipartite CV entanglement in thesystem. In particular, by increasing the LDP, one cansee that the field-atom and atom-mirror entanglementincrease at the expense of optical-mechanical entangle-ment. The intracavity mode is able to mediate for therealization of a robust stationary (i.e., persistent) entan-glement between the MR mode and the single two-levelatom, which could be extremely useful in quantum in-formation/quantum computer networks in which the MRmodes are used for quantum communications[67, 68], andthe atom is used as a qubit(e.g., solid-state qubits). Fur-thermore, we have analyzed the occurrence of the NMS inthe displacement spectrum of the oscillating mirror. Aswe have shown, for a small value of the LDP, the usualnormal-mode splitting into two modes with central peaksat the sidebands ω = ± ω m is observed and by increasingthe LDP the normal mode splits up into three modes. Wehave shown that, when the LDP is large enough the me-chanical mode, the photon mode, and the atomic modeforms a system of three coupled oscillators. The realiza-tion of such a scheme will also open new opportunities for the implementation of quantum teleportation and/or thephoton blockade process to prevent two or more photonsfrom entering the cavity at the same time. ACKNOWLEDGEMENTS
The authors wish to thank The Office of GraduateStudies of The University of Isfahan for their support.
Appendix: Definition of the elements of the driftmatrix of Eq.(29)
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