Abstract
We have explored steady-state entanglement and steering in a cross-shaped double-cavity with a levitating magnetic sphere and driven by a squeezed vacuum field. From the equations of motion of the system, we derived the linearized Lyapunov equation for the steady-states, introduced the logarithmic negativity to measure the entanglement and a similar quantity to measure the steering, and conducted calculations in parameter regions around stable steady-states. Numerical results show that steady-state entanglement and steering between various components of the system, such as cavity-cavity, cavity-magnon, and cavity-phonon, can form by choosing experimentally feasible detunings, dissipation rates, and coupling rates. Moreover, one-way steering can be achieved through appropriately adjusting the system parameters. The proposed model provides a platform for the study of the entanglement and one-way steering, and may find itself applications in quantum information processing and quantum cryptography.
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References
Amazioug, Mohamed, Singh, Shailendra: Berihu Teklu and and Muhammad Asjad, Feedback Control of Quantum Correlations in a Cavity Magnomechanical System with Magnon Squeezing. Entropy 25, 1462 (2023)
Bell, J.S.: On the Einstein-Podolsky-Rosen Paradox. Physics 1, 195–200 (1964)
Bemani, F., Motazedifard, A., Roknizadeh, R., Naderi, M.H., Vitali, D.: Synchronization dynamics of two nanomechanical membranes within a fabry-perot cavity Phys. Rev. A 96, 023805 (2017)
Bemani, F., Roknizadeh, R., Motazedifard, A., Naderi, M.H., Vitali, D.: Quantum correlations in optomechanical crystals. Phys. Rev. A 99, 063814 (2019)
Curty, M., Lewenstein, M., Lütkenhaus, N.: Entanglement as a Precondition for Secure Quantum Key Distribution. Phys. Rev. Lett. 92, 217903 (2004)
DeJesus, E.X., Kaufman, C.: Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. Phys. Rev. A 35, 5288 (1987)
Ebrahimi, M.S., Motazedifard, A., Harouni, M.B.: Single-quadrature quantum magnetometry in cavity electromagnonics. Phys. Rev. A 103, 062605 (2021)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)
Gardiner, C.W., Zoller, P.: Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics. Springer, Berlin, Heidelberg (2004)
Giampaolo, S.M., Illuminati, F.: Long-distance entanglement and quantum teleportation in coupled-cavity arrays. Phys. Rev. A 80, 050301 (2009)
He, Q.Y., Ficek, Z.: Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system. Phys. Rev. A 89, 74–79 (2014)
He, Q.Y., Gong, Q.H., Reid, M.D.: Classifying directional gaussian entanglement, Einstein-Podolsky-Rosen steering, and discord. Phys. Rev. Lett. 114, 060402 (2015)
Hidki, Abdelkader, Ren, Ya-Long., Lakhfif, Abderrahim, El Qars, Jamal, Nassik, Mostafa: Enhanced maximum entanglement between two microwave fields in the cavity magnomechanics with an optical parametric amplifier. Phys. Lett. A 463, 128667 (2023)
Hirota, O., Holevo, A.S., Caves, C.M.: Quantum communication, computing, and measurement. Springer (2012)
Honjo, T., Nam, S.W., Takesue, H., Zhang, Q., Kamada, H., Nishida, Y., Tadanaga, O., Asobe, M., Baek, B., Hadfield, R., Miki, S., Fujiwara, M., Sasaki, M., Wang, Z., Inoue, K., Yamamoto, Y.: Long-distance entanglement-based quantum key distribution over optical fiber. Opt. Exp. 16, 19118–19126 (2008)
Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and quantum communication. Springer, Berlin Heidelberg (2001)
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Hu, C.Y., Rarity, J.G.: Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity. Phys. Rev. B 83, 115303 (2011)
Karlsson, A., Koashi, M., Imoto, N.: Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A 59, 162 (1999)
Li, Jie, Groblacher, Simon: Entangling the vibrational modes of two massive ferromagnetic spheres using cavity magnomechanics. Quant. Sci. Technol. 6, 02405 (2021)
Linden, N., Popescu, S.: Good Dynamics versus Bad Kinematics: Is Entanglement Needed for Quantum Computation? Phys. Rev. Lett. 87, 047901 (2001)
Li, J., Zhu, S.Y.: Entangling two magnon modes via magnetostrictive interaction. New J. Phys. 21, 085001 (2019)
Li, J., Zhu, S.-Y., Agarwal, G.S.: Squeezed states of magnons and phonons in cavity magnomechanics. Phys. Rev. A 99, 021801 (2019)
Li, J., Zhu, S.-Y., Agarwal, G.S.: Magnon-photon-phonon entanglement in cavity magnomechanics. Phys. Rev. Lett. 121, 203601 (2018)
Maity, A.G., Datta, S., Majumdar, A.S.: Tighter Einstein-Podolsky-Rosen steering inequality based on the sum-uncertainty relation. Phys. Rev. A 96, 052326 (2017)
Midgley, S.L.W., Ferris, A.J., Olsen, M.K.: Asymmetric Gaussian steering: when Alice and Bob disagree. Phys. Rev. A 81, 022101 (2010)
Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press (2010)
Rigas, J., Gühne, O., Lütkenhaus, N.: Entanglement verification for quantum-key-distribution systems with an underlying bipartite qubit-mode structure. Phys. Rev. A 73, 012341 (2006)
Schrödinger, E.: Discussion of probability relations between separated systems. Math. Pro. Cambridge 31, 555–563 (1935)
Schrödinger, E.: Probability relations between separated systems. Math. Pro. Cambridge 32, 446 (1936)
Serga, A.A., Chumak, A.V., Hillebrands, B.: Magnonic crystals for data processing. J. Phys. D Appl. Phys. 43, 264002 (2010)
Solki, H., Motazedifard, A., Naderi, M.H.: Improving photon blockade, entanglement, and mechanical-cat-state generation in a generalized cross-kerr optomechanical circuit. Phys. Rev. A 108, 063505 (2023)
Tan, Huatang: Genuine photon-magnon-phonon Einstein-Podolsky-Rosen steerable nonlocality in a continuously-monitored cavity magnomechanical syste. Phys. Rev. Res. 1, 033161 (2019)
Uola, R., Acs, C., Nguyen, H.C.: Quantum steering. Rev. Mod. Phys. 92, 1 (2020)
Vitali, D., Gigan, S., Ferreira, A., Böhm, H.R., Tombesi, P., Guerreiro, A., Vedral, V., Zeilinger, A., Aspelmeyer, M.: Optomechanical entanglement between a movable mirror and a cavity field. Phys. Rev. Lett. 98, 030405 (2007)
Xie, H., He, L., Liao, Chang-Geng., Chen, Zhi-Hua., Lin, Xiu-Min.: Generation of robust optical entanglement in cavity optomagnonics. Opt. Express 31, 7994–8004 (2023)
Yang, Zhi-Bo., Liu, Xuan-De., Yin, Xin-Yi., Ming, Ying, Liu, Hong-Yu., Yang, Rong-Can.: Controlling stationary one-way quantum steering in cavity magnonics. Phys. Rev. Appl. 15, 024042 (2021)
Yuen, H., Shapiro, J.: Optical communication with two-photon coherent states- part i: quantum-state propagation and quantum-noise reduction. IEEE Trans. Inf. Theory 24, 657–668 (1978)
Zhang, X., Zou, C.L., Jiang, L., Tang, H.X.: Cavity magnomechanics. Sci. Adv. 2, 1501286 (2016)
Zhang, D., Wang, X.M., Li, T.F., Luo, X.Q., Wu, W., Nori, F., You, J.: Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere. NPJ Quant. Inf. 1, 15014 (2015)
Zhang, Wei, Wang, Tie, Han, Xue, Zhang, Shou, Wang, Hong-Fu.: Quantum entanglement and one-way steering in a cavity magnomechanical system via a squeezed vacuum field. Opt. Exp. 15, 10969–10980 (2021)
Zhao, Yabo, Zhao, Ruiqing, Chen, Lanxin, Pan, Jingyu, Zhang, Mei: Steady-state entanglement in a mechanically coupled double cavity containing magnetic spheres. Quant. Inf. Process. 22, 307 (2022)
Acknowledgements
M. Z. thanks Junzhong Yang for helpful discussions. This work was supported by the National Natural Science Foundation of China under Grant No. 11475021 and the National Key Basic Research Program of China under Grant No. 2013CB922000.
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R.Z. wrote the main manuscript text and prepared all the figures. J.J. and L.W. reviewed the manuscript. M.Z. drafted the manuscript.
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Zhao, R., Jia, J., Wu, L. et al. Entanglement and steering in a cross-shaped double-cavity with a magnetic sphere and driven by a squeezed vacuum field. Opt Quant Electron 56, 947 (2024). https://doi.org/10.1007/s11082-024-06859-w
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DOI: https://doi.org/10.1007/s11082-024-06859-w