Macroscopic Entanglement of Remote Optomechanical Systems Assisted by Parametric Interactions

Transcription

1 Int J Theor Phys ( : DOI /s Macroscopic Entanglement of Remote Optomechanical Systems Assisted by Parametric Interactions Yong-Hong Ma E Wu Received: 21 May 2014 / Accepted: 5 September 2014 / Published online: 24 September 2014 Springer Science+Business Media New York 2014 Abstract A scheme to generate stationary entanglement between two distant Fabry-Perot cavities with parametric interactions is proposed in the current paper. The bipartite entanglement between two distant mirrors and the entanglement between a mirror and a distant cavity mode are quantified. The logarithmic negativity, which characterizes continuousvariable (CV entanglement between two systems, is found to increase with the help of parametric interactions. Moreover, such macroscopic entanglement between two distant mirrors and between a mirror and a distant cavity mode persists for environment temperatures of about 60 K and 6 K, respectively. Keywords Macroscopic entanglement Optomechanical systems Parametric interactions 1 Introduction Entanglement, central to quantum mechanics, has become a important resource for many quantum information processes [1]. So far, theoretical andexperimental endeavorhave been directed towards preparation and manipulation in microscopic systems, comprised by such as photons, ions, and atoms [2 10]. Many scientists are earnest desire to observe the possibilities of obtaining quantum entanglement at the macroscopic level. Over the last decades, Y.-H. Ma ( Key Laboratory of Integrated Exploitation of Bayan Obo Multi-Metal Resources, Inner Mongolia University of Science and Technology, Baotou , China myh dlut@126.com Y.-H. Ma EWu School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou , China

2 Int J Theor Phys ( : thanks to the theoretical study and technological progress, it has now been possible to obtain entanglement in mesoscopic and even macroscopic systems. With the fast development of the ground-state cooling of micromechanical systems [11 14], optomechanical systems with high resonance and large quality factors have recently become important candidates in macroscopics systems [15 20]. In recent years, various forms to generate bipartite entanglements have been proposed. Such as, the entanglement between two mirrors have been proposed in a optomechancal systems [21 26], and the optomechanical entanglement between a cavity field and a movable mirror have been studied in Refs. [27 30]. These bipartite entanglements are generated in an optomechanical system including the entanglement between a cavity and a near mirror or the entanglement between two closer mirrors. So far, there has been much less work on the generation of macroscopic entanglement in distant optomechanical system. As far as we know, there is only one scheme proposed for distant optomechanical entanglement [31], in which a setup is comprised by two identical cavities which are coupled by an optical fiber. In their numerical calculations, they give time evolution of the degree of entanglement as a function of the effective detuning, the parameter ga s = gs b = 2.5 is behaved as a constant. However, they have derived ga s = gs b = iλη+η(k+i,that λ 2 +κ 2 +2i κ 2 is, the parameter ga s = gs b is the function of the effective detuning. Obviously, it is not reasonable that ga s = gs b is considered to be a constant. Based on the Ref. [31], we consider two distant Fabry-Perot cavities coupled by an optical fiber, each cavity with one movable mirror and a fixed mirror. Different with previous work [31], two degenerate optical parametric amplifiers (OPA are embedded between the moveable mirror and fixed mirror of each cavity. This exploration is motivated by the following reasons. Firstly, when an OPA is added inside a cavity, the optomechanical coupling, the normal mode splitting, and the cooling of the mechanical mirror can be considerably improved [32]. Secondly, it has been proved that, by tuning OPA in an optomechanical cavity, the cavity energy shift is used to reduce the photon number fluctuation [33]. Thus, based on these reasons, when the OPA is embodied into the cavity with a mechanical oscillator, whether the degree of the entanglement can be enhanced? The feasible experimental parameters are from current experiments on optomechanical normal mode splitting [34, 35]. We investigate the entanglement between two distant mirrors and the entanglement between a mirror and a distant cavity mode. The results show, with the help of OPA, the maximally entangled value E N which characterised the entanglement between a mirror and a distant cavity mode can be increased. The entanglement between two distant mirrors and the entanglement between a mirror and a distant cavity field are both very robust, and they persist for environmental temperatures above 60 K and 6 K, respectively. The entanglement of such remote mechanical elements is of importance for constructing long-distance quantum communication networks [36]. 2 Model and Hamiltonian of the System We consider a scheme sketched in Fig. 1 under the present study: two Fabry-Perot cavities each with one fixed mirror and one perfectly reflective movable mirror are coupled to each other by an optical fiber. Two nonlinear crystals labeled OPA in Fig. 1 are embedded between the moveable mirror and fixed mirror of each cavity. Two external lasers with the same frequencies ω 0 are injected into two cavities through two fixed mirrors. Two movable mirrors are treated as quantum-mechanical oscillators with the same effective mass m

3 1336 Int J Theor Phys ( : M1 OPA OPA Optical fiber coupling M2 ω 0 Fig. 1 Sketch of the system: two OPAs are injected into two Fabry-Perot cavities with two fixed mirrors and two oscillating mirrors. Two Fabry-Perot cavities are coupled to each other by an optical fiber and frequency ω m. In the interaction picture, the Hamiltonian of this system reads as follows: H ( = (ω c ω 0 a 1 a 1 + a 2 a ω m +i ε j (a j a j + i ( G j j=1,2 j=1,2 j=1,2 ( ( pj 2 + q2 j + λ a 1 a 2 + a 2 a 1 e iθ a 2 e iθ aj 2 j j=1,2 χ j a j a j q j, (1 where a j and a j are the creation and annihilation operators of the cavity modes. We assume two cavities have the same frequency ω c and decay rate κ. In(1 the first term denotes the free Hamiltonian of two cavities. The second term are the energies of the mechanical oscillators with the positions and momentums operators q j and p j (j = 1, 2. The third term denotes the coupling between two cavities by an optical fiber with intermode coupling constant λ. The fourth term describes the radiation-pressure coupling with the coupling rate χ j (j = 1, 2. The fifth term describes the input lasers, and the parameter ε 1 = ε 2 = ε is related to the input laser power P with ε = 2Pκ/ ω 0. The last term arises from nonlinear crystals, here we assume the parameter G 1 = G 2 = G is proportional to the pump driving the OPA. By the Heisenberg equations of motion and including the effect of damping and noises, the following quantum Langevin equations can be obtained: q i = ω m p i, ṗ i = ω m q i γ m p i + χa i a i + ξ i (i = 1, 2, a j = (κ + iω c iω 0 a j + iχa j q j iλa k + ε + 2Ge iθ a j + 2κa jin (j, k = 1, 2, (2 the two movable mirrors are damped at the same damping rates γ m. Meanwhile, the vacuum radiation input noises are expressed as x jin = a jin +a jin,andy jin = i (a jin a jin (j = 1, 2, in 2 2 which the nonzero correlation function is [37] a jin (ta jin (t =δ(t t, and the thermal Langevin force ξ j is resulting from the coupling of the movable mirrors to the environment, with the following correlation function [38]: ξ j (tξ j (t = γ m (t t ωe iω [ ( ] ω 1 + coth dω, (3 2πω m 2k B T where k B is the Boltzmann constant and T is the temperature of the environment. From (2 and (3, the steady-state mean values can be derived, such as p1 s = ps 2 = 0, αs = α1 s = αs 2 = ε[κ+2g cos θ i( +λ 2G sin θ], q s κ λ 2 +2 λ 4G 2 1 = qs 2 = χ αs 2 ω m, The effective cavity detuning, including

4 Int J Theor Phys ( : radiation pressure effects, can be expressed as = ω c ω 0 χ 2 α s 2 /ω m. The linearized Langevin equations thus takes the form δq i = ω m δp i, δṗ i = γ m δp i ω m δq i + G s δx i + ξ i,(i = 1, 2 δx i = (κ 2G cos θδx i + ( + 2G sin θδy i + λδy j + 2κx iin, (4 δy j = (κ + 2G cos θδy j ( + 2G sin θδx j λδx k + G s δq j + 2κy jin,(j,k = 1, 2 with G s = 2χα s.(4 can be rewritten as the matrix form f(t= Af (t + χ(t, (5 in which the transposes of the column vectors f(t and χ(t can be expressed as f(t T = (δq 1,δp 1,δq 2,δp 2,δx 1,δy 1,δx 2,δy 2 and χ(t T = (0,ξ 1, 0,ξ 2, 2κδx 1in, 2κδy1in, 2κδx 2in, 2κδy 2in, respectively. The drift matrix A for the above equation is given by 0 ω m ω m γ m 0 0 G s ω m A= 0 0 ω m γ m 0 0 G s k + 2G cos θ + 2G sin θ 0 λ, (6 G s ( 2G sin θ (k + 2G cos θ λ λ k + 2G cos θ + 2G sin θ 0 0 G s 0 λ 0 ( 2G sin θ (k + 2G cos θ where n = (exp{ ω m /k B T } 1 1 refers to the average excitation number. The purpose of the present paper is to study the entanglement properties in the steady state, which can be guaranteed to exist if the real parts of the eigenvalues of A are negative. The detailed conditions for this condition to occur are derived by a simple stability-analysis test [39]. 3 The Entanglement of the Distant Optomechanical Systems When the steady-state conditions are satisfied, the following equation for the stability correlation matrix can be obtained AV + VA T = D (7 with D = Diag [0,γ m (2n + 1, 0,γ m (2n + 1, κ, κ, κ, κ], in(7 V ij = ( u i ( u j ( + u j ( u i ( /2, with u T ( = (δq 1 (, δp 1 (, δq 2 (, δp 2 (, δx 1 (, δy 1 (, δx 2 (, δy 2 ( is the vector of the fluctuation operators at the steady state (t. The behavior of the CV entanglement between a movable mirror and a distant cavity field and the entanglement between two distant mirrors will be investigated. As the Langevin (2 is linear, the initial Gaussian state of the system is preserved. We employ the logarithmic negativity E N [40 43] to ascertain the entanglement. In the CV case, E N is defined as [42] E N = max[0, log 2 ν ], (8

5 1338 Int J Theor Phys ( : where the smallest eigenvalue ν = D 2detC, and the correlation matrix Ɣ is Ɣ = (Ɣ (Ɣ 2 4DetƔ 2, with (Ɣ = det B + det ( B C C T D The elements of the covariance matrix Ɣ ij is the 4 4 matrix from V which comes from (7. Based on current experiment conditions [44 46], the bipartite CV entanglement in the cavity field-mirror subsystem will be described and quantified. Firstly, we consider the entanglement between a movable mirror and adjacent cavity field. In this case, we need only consider either optomechanical system of two cavities. If the parameter satisfy λ = 0,G = 0, this scheme will return to the model of [27] investigated. Take the same parameters with [27], our numerical simulations are the same with previous (9 Fig. 2 The logarithmic negativity E N as a function of normalized detuning /ω m and the nonlinear parameter G, which describe the entanglement between two distant mirrors (a and the entanglement between a mirror and a distant cavity field (b. The choice of the parameters are: ω m = 20π MHz; γ m = 200π Hz; L = 1 mm; m = 5 ng; finesse F = , and driven by a laser with wavelength 810 nm, power P = 50 mw; θ = 45, T = 0.2 K

6 Int J Theor Phys ( : (see Fig. 1 and 2 of [27]. In this paper, we are interested in the entanglement between distant optomechanical systems. In Fig 2 the logarithmic negativity E N is plotted as a function of the normalized detuning /ω m and the nonlinear parameter G which is related to the OPAs. As is clear from Fig. 2a, when the OPAs have not added into the optomechanical systems (corresponding to G = 0, we find that the entanglement between two distant mirrors exists for effective detuning from about 0.75 ω m to 2 ω m.thevaluee N is closest to its maximum 0.55 at about 1.5 /ω m. With the help of OPAs (corresponding to G = 0, the maximum E N leaves from 1.5 /ω m to about /ω m, and the maximum E N can be increased from 0.5 to 1.9 (corresponding to G = More importantly, under the action of cavity-opa coupling, the range of the entanglement between two distant mirrors can be broaden. More broader of effective detuning is obtained, the more easily it is realized in experiment. Fig. 3 The logarithmic negativity, described the entanglement between two distant movable mirrors (a and the entanglement between a movable mirror and a distant cavity mode (b, as a function of the nonlinear parameter G and the temperature environment T. The choice of the parameters are: ω m = 20π MHz; γ m = 200π Hz; L = 1 mm; m = 5 ng; finesse F = , /ω m = 1.2 and driven by a laser with wavelength 810 nm, power P = 50 mw; θ = 45

7 1340 Int J Theor Phys ( : Figure 2b shows CV entanglement between a mirror and a distant cavity field changes with the normalized detuning /ω m and the nonlinear parameter G. When two OPAs have not been added into two cavities (G = 0, it shows the entanglement between a mirror and a distant cavity field barely takes on normal-mode splitting. With the parameter G increasing, the maximally entangled value E N can be greatly enhance from about 0.51 (G = 0 to 2.4 (G = It can be account for: OPAs in the cavities cause a strong coupling between the movable mirror and the cavity field. Moreover, the width of the curve is enhanced with the help of OPAs. To investigate the robustness of this distant optomechanical entanglement, in Fig. 3 the robustness of such distant CV entanglement dependent on the environmental temperature is presented. As the temperature of the environment increase, the degree of such distant optomechanical entanglement between two distant mirrors and the entanglement between a mirror and a distant cavity field both decreases due to the thermal fluctuations. However, with the present of two OPAs, such distant optomechanical entanglement between two distant mirrors and the entanglement between a mirror and a distant cavity field are both very robust, and the persistence for environmental temperatures can be extended to above 60 K and 6 K, respectively. So the lower environment temperature can make the entanglement between two distant mirrors and the entanglement between a mirror and a distant cavity field are both stronger. 4 Conclusions In conclusion, a scheme to generate a stationary macroscopic entanglement between two distant Fabry-Perot cavities is presented in the current study. Firstly, the CV entanglement between two distant mirrors in this system is studied. It shws that when OPAs add into two cavities, the maximum entanglement between two distant mirrors leaves from 1.5 /ω m to about /ω m, and the maximum E N can be increased from 0.5 to 1.9, and the range of the entanglement between two distant mirrors can be broaden. Secondly, we investigate the entanglement between a movable mirror and a distant cavity field in this system. Because adding OPAs into cavities cause a strong coupling between a movable mirror and the cavity field, the maximally entangled value E N which characterised the entanglement between a mirror and a distant cavity mode can be increased from 0.51 to 2.4. On the other hand, the width of the curve is greatly enhanced. Lastly, we consider such distant optomechanical entanglement dependent of environment temperatures. The results show that the entanglement between two distant mirrors and between a mirror and a distant cavity field are both very robust, and they persist for environmental temperatures above 60 K and 6 K, respectively. As a result, with the presence of OPAs, the stronger distant entanglement with high environment temperature can be created, this advantage will be more conducive for practical applications. The entanglement of this remote mechanical system will be a very important resources for constructing long-distance quantum communication networks. Acknowledgments Project supported by the National Natural Science Foundation of China (Grant No and , and Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grants No. NJYT-13-B13,and the Key Project of Chinese Ministry of Education (Grant No

8 Int J Theor Phys ( : References 1. Nielson, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University ( Zheng, S.B.: Phys. Rev. A 89, ( Zheng, S.B.: Phys. Rev. A 87, ( Xia, Y., Song, J., Lu, P.M., Song, H.S.: J. Phys. B At. Mol. Opt. Phys. 44, ( Shi, Z.C., Xia, Y., Song, J., Song, H.S.: Quan. Inf. Comput. 12, ( Xia, Y., Song, H.S.: Phys. Lett. A 364, 117 ( Yang G.H., Zhou, L.: Eur. Phys. J. D 47, 277 ( Ma, Y.H., Mu, Q.X., Zhou, L.: J. Phys. B At. Mol. Opt. Phys. 41, ( Esteve, D., Raimond, J.M., Dalibard, J.: Quantum Entanlement and Information Processing. Cambridge University ( Horodecki, R., et al.: Rev. Mod. Phys. 81, ( Cole, D., Gigan, S.K., Schwab, C.S., Aspelmeyer, M.: Nat. Phys. 5, 485 ( Verhagen, E., Deleglise, S., Weis, S., Schliesser, A., Kippenberg, T.J.: Nat. Lond. 482, 63 ( Teufel, J.D., et al.: Nat. Lond. 475, 359 ( Teufel, J.D., et al.: Nature 471, 204 ( Huang, S., Agarwal, G.S.: Phys. Rev. A 83, ( Huang, S., Agarwal, G.S.: Phys. Rev. A 80, ( Rocheleau, T., Ndukum, T., Macklin, C., Hertzberg, J.B., Clerk, A.A., Achwab, K.C.: Nature 463, ( O Connell, A.D., et al.: Nature 464, ( Brown, K.R., et al.: Nature 471, ( Zhao, C., et al.: Phys. Rev. Lett. 102, ( Mancini, S., Giovannetti, V., Vitali, D., Tombesi, P.: Phys. Rev. Lett. 88, ( Zhang, J., Peng, K., Braunstein, S.L.: Phys. Rev. A 68, ( M. Pinard, et al.: Lett. Europhnys. 72, 747 ( Pinard, M., Dantan, A., Vitali, D., Arcizet, O., Briant, T., Heidmann, A.: Europhys. Lett. 72, 747 ( Vitali D, Mancini S, Tombesi, P.: J. Phys. A: Math. Theor. 40, 8055 ( Huang, S.M., Agarwal, G.S.: New. J. Phys. 11, ( Vitali, D., Gigan, S., Ferreira, A., Böhm, H.R., Tombesi, P., Guerreiro, A., Vedral, V., Zeilinger, A., Aspelmeyer, M.: Phys. Rev. Lett. 98, ( Genes, C., Mari, A., Tombesi, P., Vitali, D.: Phys. Rev. A 78, ( Ma, Y.H., Zhou, L.: J. Appl. Phys. 111, ( Zhang, D., Zhang, X.P., Zheng Qiang, Q.: Chinese Phys. B 22, ( Joshi, C., Larson, J., Jonson, M., Anderson, E., öhberg, P.: Phys. Rev. A 85, ( Huang, S., Agarwal, G.S.: Phys. Rev. A 79, ( Kumar, T., Bhattacherjee, A.B., ManMohan, P.hys.: Rev. A 81, ( Gröblacher, S., Hammerer, K., Vanner, M.R., Aspelmeyer, M.: Nat. Lond. 460, 724 ( Gorodetsky, M.L., Schliesser, A., Anetsberger, G., Deleglise, S., Kippenberg, T.J.: Opt. Express 18, ( Kimble, H.J.: Nature 453, 1023 ( Gardiner, C.W., Zoller, P.: Quantum Noise. Springer-Verlag, Berlin ( Giovannetti, V., Vitali, D.: Phys. Rev. A 63, ( DeJesus, E.X., Kaufman, C.: Phys. Rev. A 35, ( Laurat, J., Keller, G., Augusto Oliveira-Huguenin, J., Fabre, C., Coudreau, T., Serafini, A., Adesso, G., Illuminati, F.: J. Opt. B: Quantum Semiclass. Opt. 7, S577 S587 ( Vidal, G., Werner, R.F.: Pyhs. Rew. A 65, ( Adesso, G., Serafini, A., Illuminati, F.: Phys. Rev. A 70, ( Ma, Y.H., Mu, Q.X., Yang, G.H., Zhou, L.: J. Phys. B: At. Mol. Opt. Phys. 41, ( Gigan, S., Böhm, H.R., Paternostro, M., Blaser, F., Langer, G., Hertzberg, J.B., Schwab, K.C., Bäuerle, D., Aspelmeyer, M., Zeilinger, A.: Nat. Lond. 444, ( Kleckner, D., Bouwmeester, D.: Nat. Lond. 444, ( Marquardt, F., Chen, J.P., Clerk, A.A., Girvin, S.M.: Phys. Rev. Lett. 99, (2007