Abstract

We propose a scheme to generate strong squeezing of a mechanical oscillator in an optomechanical system through Lyapunov control. Frequency modulation of the mechanical oscillator is designed via Lyapunov control. We show that the momentum variance of the mechanical oscillator decreases with time evolution in a weak coupling case. As a result, strong mechanical squeezing is realized quickly (beyond 3 dB). In addition, the proposal is immune to cavity decay. Moreover, we show that the obtained squeezing can be detected via an ancillary cavity mode with homodyne detection.

© 2020 Chinese Laser Press

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    [Crossref]
  2. C.-S. Hu, Z.-B. Yang, H. Wu, Y. Li, and S.-B. Zheng, “Twofold mechanical squeezing in a cavity optomechanical system,” Phys. Rev. A 98, 023807 (2018).
    [Crossref]
  3. W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36–44 (1991).
    [Crossref]
  4. S.-L. Ma, X.-K. Li, J.-K. Xie, and F.-L. Li, “Two-mode squeezed states of two separated nitrogen-vacancy-center ensembles coupled via dissipative photons of superconducting resonators,” Phys. Rev. A 99, 012325 (2019).
    [Crossref]
  5. V. Peano, H. G. L. Schwefel, C. Marquardt, and F. Marquardt, “Intracavity squeezing can enhance quantum-limited optomechanical position detection through deamplification,” Phys. Rev. Lett. 115, 243603 (2015).
    [Crossref]
  6. B. Xie and S. Feng, “Squeezing-enhanced heterodyne detection of 10 Hz atto-Watt optical signals,” Opt. Lett. 43, 6073–6076 (2018).
    [Crossref]
  7. A. Motazedifard, F. Bemani, M. H. Naderi, R. Roknizadeh, and D. Vitali, “Force sensing based on coherent quantum noise cancellation in a hybrid optomechanical cavity with squeezed-vacuum injection,” New J. Phys. 18, 073040 (2016).
    [Crossref]
  8. J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams, T. Adams, P. Addesso, and R. X. Adhikari, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
    [Crossref]
  9. M. A. Lemonde, N. Didier, and A. A. Clerk, “Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification,” Nat. Commun. 7, 11338 (2016).
    [Crossref]
  10. Y. Wang, C. Li, E. M. Sampuli, J. Song, Y. Jiang, and Y. Xia, “Enhancement of coherent dipole coupling between two atoms via squeezing a cavity mode,” Phys. Rev. A 99, 023833 (2019).
    [Crossref]
  11. X.-Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
    [Crossref]
  12. D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67, 699–702 (1991).
    [Crossref]
  13. W. Ge and M. Bhattacharya, “Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence,” New J. Phys. 18, 103002 (2016).
    [Crossref]
  14. A. Serafini, A. Retzker, and M. B. Plenio, “Generation of continuous variable squeezing and entanglement of trapped ions in time-varying potentials,” Quantum Inform. Process. 8, 619 (2009).
    [Crossref]
  15. W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-Markovian regime,” New J. Phys. 19, 083022 (2017).
    [Crossref]
  16. J. Liu and K.-D. Zhu, “Coupled quantum molecular cavity optomechanics with surface plasmon enhancement,” Photon. Res. 5, 450–456 (2017).
    [Crossref]
  17. B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Ann. Phys. 385, 757–768 (2017).
    [Crossref]
  18. J. Liu and K.-D. Zhu, “Room temperature optical mass sensor with an artificial molecular structure based on surface plasmon optomechanics,” Photon. Res. 6, 867–874 (2018).
    [Crossref]
  19. A. Motazedifard, A. Dalafi, M. Naderi, and R. Roknizadeh, “Strong quadrature squeezing and quantum amplification in a coupled Bose-Einstein condensate-optomechanical cavity based on parametric modulation,” Ann. Phys. 405, 202–219 (2019).
    [Crossref]
  20. B. A. Levitan, A. Metelmann, and A. A. Clerk, “Optomechanics with two-phonon driving,” New J. Phys. 18, 093014 (2016).
    [Crossref]
  21. X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
    [Crossref]
  22. J.-Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
    [Crossref]
  23. C.-H. Bai, D.-Y. Wang, S. Zhang, and H.-F. Wang, “Qubit-assisted squeezing of mirror motion in a dissipative cavity optomechanical system,” Sci. China Phys. Mech. Astron. 62, 970311 (2019).
    [Crossref]
  24. B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Optomechanical quadrature squeezing in the non-Markovian regime,” Opt. Lett. 43, 6053–6056 (2018).
    [Crossref]
  25. Z.-C. Zhang, Y.-P. Wang, Y.-F. Yu, and Z.-M. Zhang, “Quantum squeezing in a modulated optomechanical system,” Opt. Express 26, 11915–11927 (2018).
    [Crossref]
  26. M. Rashid, T. Tufarelli, J. Bateman, J. Vovrosh, D. Hempston, M. S. Kim, and H. Ulbricht, “Experimental realization of a thermal squeezed state of levitated optomechanics,” Phys. Rev. Lett. 117, 273601 (2016).
    [Crossref]
  27. D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
    [Crossref]
  28. G. Milburn and D. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401–404 (1981).
    [Crossref]
  29. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  30. G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
    [Crossref]
  31. K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009).
    [Crossref]
  32. A. Dalafi, M. H. Naderi, and A. Motazedifard, “Effects of quadratic coupling and squeezed vacuum injection in an optomechanical cavity assisted with a Bose-Einstein condensate,” Phys. Rev. A 97, 043619 (2018).
    [Crossref]
  33. M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89, 023849 (2014).
    [Crossref]
  34. C.-H. Bai, D.-Y. Wang, S. Zhang, S. Liu, and H.-F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photon. Res. 7, 1229–1239 (2019).
    [Crossref]
  35. A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, “Mechanical squeezing via parametric amplification and weak measurement,” Phys. Rev. Lett. 107, 213603 (2011).
    [Crossref]
  36. A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).
    [Crossref]
  37. C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum nondemolition measurement of a quantum squeezed state beyond the 3 dB limit,” Phys. Rev. Lett. 117, 100801 (2016).
    [Crossref]
  38. X. You, Z. Li, and Y. Li, “Strong quantum squeezing of mechanical resonator via parametric amplification and coherent feedback,” Phys. Rev. A 96, 063811 (2017).
    [Crossref]
  39. R. Zhang, Y. Fang, Y.-Y. Wang, S. Chesi, and Y.-D. Wang, “Strong mechanical squeezing in an unresolved-sideband optomechanical system,” Phys. Rev. A 99, 043805 (2019).
    [Crossref]
  40. J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115, 243601 (2015).
    [Crossref]
  41. W.-J. Gu, Z. Yi, L.-H. Sun, and Y. Yan, “Generation of mechanical squeezing and entanglement via mechanical modulations,” Opt. Express 26, 30773–30785 (2018).
    [Crossref]
  42. C. Li, J. Song, Y. Xia, and W. Ding, “Driving many distant atoms into high-fidelity steady state entanglement via Lyapunov control,” Opt. Express 26, 951–962 (2018).
    [Crossref]
  43. S. Kuang and S. Cong, “Lyapunov control methods of closed quantum systems,” Automatica 44, 98–108 (2008).
    [Crossref]
  44. D. Ran, W.-J. Shan, Z.-C. Shi, Z.-B. Yang, J. Song, and Y. Xia, “High fidelity Dicke-state generation with Lyapunov control in circuit QED system,” Ann. Phys. 396, 44–55 (2018).
    [Crossref]
  45. W. Li, C. Li, and H. Song, “Quantum synchronization in an optomechanical system based on Lyapunov control,” Phys. Rev. E 93, 062221 (2016).
    [Crossref]
  46. D. Ran, Z.-C. Shi, J. Song, and Y. Xia, “Speeding up adiabatic passage by adding Lyapunov control,” Phys. Rev. A 96, 033803 (2017).
    [Crossref]
  47. Z. C. Shi, L. C. Wang, and X. X. Yi, “Preparing entangled states by Lyapunov control,” Quantum Inform. Process. 15, 4939–4953 (2016).
    [Crossref]
  48. Y.-X. Zeng, T. Gebremariam, M.-S. Ding, and C. Li, “Quantum optical diode based on Lyapunov control in a superconducting system,” J. Opt. Soc. Am. B 35, 2334–2341 (2018).
    [Crossref]
  49. W.-M. Zhang, K.-M. Hu, Z.-K. Peng, and G. Meng, “Tunable micro- and nanomechanical resonators,” Sensors 15, 26478–26566 (2015).
    [Crossref]
  50. D.-Y. Wang, C.-H. Bai, S. Liu, S. Zhang, and H.-F. Wang, “Optomechanical cooling beyond the quantum backaction limit with frequency modulation,” Phys. Rev. A 98, 023816 (2018).
    [Crossref]
  51. A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
    [Crossref]
  52. R. A. Barton, I. R. Storch, V. P. Adiga, R. Sakakibara, B. R. Cipriany, B. Ilic, S. P. Wang, P. Ong, P. L. McEuen, J. M. Parpia, and H. G. Craighead, “Photothermal self-oscillation and laser cooling of graphene optomechanical systems,” Nano Lett. 12, 4681–4686 (2012).
    [Crossref]
  53. C. Chen, S. Lee, V. V. Deshpande, G.-H. Lee, M. Lekas, K. Shepard, and J. Hone, “Graphene mechanical oscillators with tunable frequency,” Nat. Nanotechnol. 8, 923–927 (2013).
    [Crossref]
  54. V. Singh, S. Bosman, B. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
    [Crossref]
  55. J.-Q. Liao and C. K. Law, “Cooling of a mirror in cavity optomechanics with a chirped pulse,” Phys. Rev. A 84, 053838 (2011).
    [Crossref]
  56. Y.-D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
    [Crossref]
  57. X.-Y. Lü, J.-Q. Liao, L. Tian, and F. Nori, “Steady-state mechanical squeezing in an optomechanical system via duffing nonlinearity,” Phys. Rev. A 91, 013834 (2015).
    [Crossref]
  58. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
    [Crossref]
  59. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  60. Y.-C. Liu, Y.-F. Xiao, X. Luan, Q. Gong, and C. W. Wong, “Coupled cavities for motional ground-state cooling and strong optomechanical coupling,” Phys. Rev. A 91, 033818 (2015).
    [Crossref]

2019 (6)

S.-L. Ma, X.-K. Li, J.-K. Xie, and F.-L. Li, “Two-mode squeezed states of two separated nitrogen-vacancy-center ensembles coupled via dissipative photons of superconducting resonators,” Phys. Rev. A 99, 012325 (2019).
[Crossref]

Y. Wang, C. Li, E. M. Sampuli, J. Song, Y. Jiang, and Y. Xia, “Enhancement of coherent dipole coupling between two atoms via squeezing a cavity mode,” Phys. Rev. A 99, 023833 (2019).
[Crossref]

A. Motazedifard, A. Dalafi, M. Naderi, and R. Roknizadeh, “Strong quadrature squeezing and quantum amplification in a coupled Bose-Einstein condensate-optomechanical cavity based on parametric modulation,” Ann. Phys. 405, 202–219 (2019).
[Crossref]

C.-H. Bai, D.-Y. Wang, S. Zhang, and H.-F. Wang, “Qubit-assisted squeezing of mirror motion in a dissipative cavity optomechanical system,” Sci. China Phys. Mech. Astron. 62, 970311 (2019).
[Crossref]

C.-H. Bai, D.-Y. Wang, S. Zhang, S. Liu, and H.-F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photon. Res. 7, 1229–1239 (2019).
[Crossref]

R. Zhang, Y. Fang, Y.-Y. Wang, S. Chesi, and Y.-D. Wang, “Strong mechanical squeezing in an unresolved-sideband optomechanical system,” Phys. Rev. A 99, 043805 (2019).
[Crossref]

2018 (11)

W.-J. Gu, Z. Yi, L.-H. Sun, and Y. Yan, “Generation of mechanical squeezing and entanglement via mechanical modulations,” Opt. Express 26, 30773–30785 (2018).
[Crossref]

C. Li, J. Song, Y. Xia, and W. Ding, “Driving many distant atoms into high-fidelity steady state entanglement via Lyapunov control,” Opt. Express 26, 951–962 (2018).
[Crossref]

D. Ran, W.-J. Shan, Z.-C. Shi, Z.-B. Yang, J. Song, and Y. Xia, “High fidelity Dicke-state generation with Lyapunov control in circuit QED system,” Ann. Phys. 396, 44–55 (2018).
[Crossref]

Y.-X. Zeng, T. Gebremariam, M.-S. Ding, and C. Li, “Quantum optical diode based on Lyapunov control in a superconducting system,” J. Opt. Soc. Am. B 35, 2334–2341 (2018).
[Crossref]

D.-Y. Wang, C.-H. Bai, S. Liu, S. Zhang, and H.-F. Wang, “Optomechanical cooling beyond the quantum backaction limit with frequency modulation,” Phys. Rev. A 98, 023816 (2018).
[Crossref]

A. Dalafi, M. H. Naderi, and A. Motazedifard, “Effects of quadratic coupling and squeezed vacuum injection in an optomechanical cavity assisted with a Bose-Einstein condensate,” Phys. Rev. A 97, 043619 (2018).
[Crossref]

J. Liu and K.-D. Zhu, “Room temperature optical mass sensor with an artificial molecular structure based on surface plasmon optomechanics,” Photon. Res. 6, 867–874 (2018).
[Crossref]

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Optomechanical quadrature squeezing in the non-Markovian regime,” Opt. Lett. 43, 6053–6056 (2018).
[Crossref]

Z.-C. Zhang, Y.-P. Wang, Y.-F. Yu, and Z.-M. Zhang, “Quantum squeezing in a modulated optomechanical system,” Opt. Express 26, 11915–11927 (2018).
[Crossref]

C.-S. Hu, Z.-B. Yang, H. Wu, Y. Li, and S.-B. Zheng, “Twofold mechanical squeezing in a cavity optomechanical system,” Phys. Rev. A 98, 023807 (2018).
[Crossref]

B. Xie and S. Feng, “Squeezing-enhanced heterodyne detection of 10 Hz atto-Watt optical signals,” Opt. Lett. 43, 6073–6076 (2018).
[Crossref]

2017 (5)

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-Markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

J. Liu and K.-D. Zhu, “Coupled quantum molecular cavity optomechanics with surface plasmon enhancement,” Photon. Res. 5, 450–456 (2017).
[Crossref]

B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Ann. Phys. 385, 757–768 (2017).
[Crossref]

X. You, Z. Li, and Y. Li, “Strong quantum squeezing of mechanical resonator via parametric amplification and coherent feedback,” Phys. Rev. A 96, 063811 (2017).
[Crossref]

D. Ran, Z.-C. Shi, J. Song, and Y. Xia, “Speeding up adiabatic passage by adding Lyapunov control,” Phys. Rev. A 96, 033803 (2017).
[Crossref]

2016 (10)

Z. C. Shi, L. C. Wang, and X. X. Yi, “Preparing entangled states by Lyapunov control,” Quantum Inform. Process. 15, 4939–4953 (2016).
[Crossref]

W. Li, C. Li, and H. Song, “Quantum synchronization in an optomechanical system based on Lyapunov control,” Phys. Rev. E 93, 062221 (2016).
[Crossref]

C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum nondemolition measurement of a quantum squeezed state beyond the 3 dB limit,” Phys. Rev. Lett. 117, 100801 (2016).
[Crossref]

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

M. Rashid, T. Tufarelli, J. Bateman, J. Vovrosh, D. Hempston, M. S. Kim, and H. Ulbricht, “Experimental realization of a thermal squeezed state of levitated optomechanics,” Phys. Rev. Lett. 117, 273601 (2016).
[Crossref]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref]

A. Motazedifard, F. Bemani, M. H. Naderi, R. Roknizadeh, and D. Vitali, “Force sensing based on coherent quantum noise cancellation in a hybrid optomechanical cavity with squeezed-vacuum injection,” New J. Phys. 18, 073040 (2016).
[Crossref]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification,” Nat. Commun. 7, 11338 (2016).
[Crossref]

B. A. Levitan, A. Metelmann, and A. A. Clerk, “Optomechanics with two-phonon driving,” New J. Phys. 18, 093014 (2016).
[Crossref]

W. Ge and M. Bhattacharya, “Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence,” New J. Phys. 18, 103002 (2016).
[Crossref]

2015 (7)

X.-Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref]

E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum squeezing of motion in a mechanical resonator,” Science 349, 952–955 (2015).
[Crossref]

V. Peano, H. G. L. Schwefel, C. Marquardt, and F. Marquardt, “Intracavity squeezing can enhance quantum-limited optomechanical position detection through deamplification,” Phys. Rev. Lett. 115, 243603 (2015).
[Crossref]

J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115, 243601 (2015).
[Crossref]

W.-M. Zhang, K.-M. Hu, Z.-K. Peng, and G. Meng, “Tunable micro- and nanomechanical resonators,” Sensors 15, 26478–26566 (2015).
[Crossref]

X.-Y. Lü, J.-Q. Liao, L. Tian, and F. Nori, “Steady-state mechanical squeezing in an optomechanical system via duffing nonlinearity,” Phys. Rev. A 91, 013834 (2015).
[Crossref]

Y.-C. Liu, Y.-F. Xiao, X. Luan, Q. Gong, and C. W. Wong, “Coupled cavities for motional ground-state cooling and strong optomechanical coupling,” Phys. Rev. A 91, 033818 (2015).
[Crossref]

2014 (4)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

V. Singh, S. Bosman, B. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
[Crossref]

M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89, 023849 (2014).
[Crossref]

2013 (3)

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).
[Crossref]

J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams, T. Adams, P. Addesso, and R. X. Adhikari, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
[Crossref]

C. Chen, S. Lee, V. V. Deshpande, G.-H. Lee, M. Lekas, K. Shepard, and J. Hone, “Graphene mechanical oscillators with tunable frequency,” Nat. Nanotechnol. 8, 923–927 (2013).
[Crossref]

2012 (3)

Y.-D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
[Crossref]

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
[Crossref]

R. A. Barton, I. R. Storch, V. P. Adiga, R. Sakakibara, B. R. Cipriany, B. Ilic, S. P. Wang, P. Ong, P. L. McEuen, J. M. Parpia, and H. G. Craighead, “Photothermal self-oscillation and laser cooling of graphene optomechanical systems,” Nano Lett. 12, 4681–4686 (2012).
[Crossref]

2011 (3)

J.-Q. Liao and C. K. Law, “Cooling of a mirror in cavity optomechanics with a chirped pulse,” Phys. Rev. A 84, 053838 (2011).
[Crossref]

J.-Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[Crossref]

A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, “Mechanical squeezing via parametric amplification and weak measurement,” Phys. Rev. Lett. 107, 213603 (2011).
[Crossref]

2009 (2)

K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009).
[Crossref]

A. Serafini, A. Retzker, and M. B. Plenio, “Generation of continuous variable squeezing and entanglement of trapped ions in time-varying potentials,” Quantum Inform. Process. 8, 619 (2009).
[Crossref]

2008 (1)

S. Kuang and S. Cong, “Lyapunov control methods of closed quantum systems,” Automatica 44, 98–108 (2008).
[Crossref]

2007 (1)

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref]

1991 (2)

D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67, 699–702 (1991).
[Crossref]

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36–44 (1991).
[Crossref]

1981 (1)

G. Milburn and D. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401–404 (1981).
[Crossref]

Aasi, J.

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Ding, W.

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A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, “Mechanical squeezing via parametric amplification and weak measurement,” Phys. Rev. Lett. 107, 213603 (2011).
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W. Ge and M. Bhattacharya, “Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence,” New J. Phys. 18, 103002 (2016).
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K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009).
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M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89, 023849 (2014).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum nondemolition measurement of a quantum squeezed state beyond the 3 dB limit,” Phys. Rev. Lett. 117, 100801 (2016).
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E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum squeezing of motion in a mechanical resonator,” Science 349, 952–955 (2015).
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A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).
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S. Kuang and S. Cong, “Lyapunov control methods of closed quantum systems,” Automatica 44, 98–108 (2008).
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J.-Q. Liao and C. K. Law, “Cooling of a mirror in cavity optomechanics with a chirped pulse,” Phys. Rev. A 84, 053838 (2011).
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J.-Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
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C. Chen, S. Lee, V. V. Deshpande, G.-H. Lee, M. Lekas, K. Shepard, and J. Hone, “Graphene mechanical oscillators with tunable frequency,” Nat. Nanotechnol. 8, 923–927 (2013).
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C. Chen, S. Lee, V. V. Deshpande, G.-H. Lee, M. Lekas, K. Shepard, and J. Hone, “Graphene mechanical oscillators with tunable frequency,” Nat. Nanotechnol. 8, 923–927 (2013).
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C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum nondemolition measurement of a quantum squeezed state beyond the 3 dB limit,” Phys. Rev. Lett. 117, 100801 (2016).
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E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum squeezing of motion in a mechanical resonator,” Science 349, 952–955 (2015).
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C. Chen, S. Lee, V. V. Deshpande, G.-H. Lee, M. Lekas, K. Shepard, and J. Hone, “Graphene mechanical oscillators with tunable frequency,” Nat. Nanotechnol. 8, 923–927 (2013).
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M. A. Lemonde, N. Didier, and A. A. Clerk, “Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification,” Nat. Commun. 7, 11338 (2016).
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B. A. Levitan, A. Metelmann, and A. A. Clerk, “Optomechanics with two-phonon driving,” New J. Phys. 18, 093014 (2016).
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Y. Wang, C. Li, E. M. Sampuli, J. Song, Y. Jiang, and Y. Xia, “Enhancement of coherent dipole coupling between two atoms via squeezing a cavity mode,” Phys. Rev. A 99, 023833 (2019).
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C.-S. Hu, Z.-B. Yang, H. Wu, Y. Li, and S.-B. Zheng, “Twofold mechanical squeezing in a cavity optomechanical system,” Phys. Rev. A 98, 023807 (2018).
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J.-Q. Liao and C. K. Law, “Cooling of a mirror in cavity optomechanics with a chirped pulse,” Phys. Rev. A 84, 053838 (2011).
[Crossref]

J.-Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
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Liu, J.

Liu, S.

C.-H. Bai, D.-Y. Wang, S. Zhang, S. Liu, and H.-F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photon. Res. 7, 1229–1239 (2019).
[Crossref]

D.-Y. Wang, C.-H. Bai, S. Liu, S. Zhang, and H.-F. Wang, “Optomechanical cooling beyond the quantum backaction limit with frequency modulation,” Phys. Rev. A 98, 023816 (2018).
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Figures (6)

Fig. 1.
Fig. 1. Schematic of the considered system, where the mechanical frequency is modulated through the tuning electrode. The left setups are used to detect the obtained mechanical squeezing.
Fig. 2.
Fig. 2. (a) and (b) show time evolution of Δp2 with the time-varying frequency ωr(t) at the control case presented in (c) and (d), respectively, where Δc=ωm, κ=0.1ωm, γ=106ωm, n¯mT=0, and c=0.2.
Fig. 3.
Fig. 3. Time evolution of Δp2 at different G and Δc in (a) and (b), respectively, where (c) and (d) are the corresponding time-varying control fields. The other parameters are the same as in Fig. 2.
Fig. 4.
Fig. 4. Wigner function in units of 1/100 for ωmt=0 and ωmt=30 in (a) and (b), respectively. The parameters are the same as in Fig. 2(a).
Fig. 5.
Fig. 5. Plot of squeezing level with different cavity decay κ in (a) and thermal phonon number n¯mT in (b). (c) and (d) are the corresponding control fields. The other parameters are the same as in Fig. 2(a).
Fig. 6.
Fig. 6. Time evolution of Δq2 with detection and without detection, where Gs/ωm=0.01 and κs/ωm=0.1. The other parameters are the same as in Fig. 2(a).

Equations (42)

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H=Hres+nfn(t)Hn,
O˙=i[O,Hres+nfn(t)Hn].
V˙(t)=Tr(ρO˙)=nfn(t)i[O,Hn],
V˙(t)=O˙=cn|i[O,Hn]|2.
H=ωcaa+px22m+12mωr2(t)x2gaax+iE(eiωdtaeiwdta),
q=mωmx,p=pxmωm,
H=ωcaa+ωm2p2+ωr2(t)2ωmq2g0aaq+iE(eiωdtaeiwdta).
q˙=ωmp,p˙=ωr2(t)ωmqγp+g0aa+ξ,a˙=(κ+iΔc)a+ig0aq+E+2κain,
ain(t)ain(t)=(n¯cT+1)δ(tt),ξ(t)ξ(t)+ξ(t)ξ(t)/2=γ(2n¯mT+1)δ(tt),
q˙c=ωmpc,p˙c=ωr2(t)ωmqcγpc+g0|α|2,α˙=(κ+iΔc)α+E,
δ˙q=ωmδp,δ˙p=ωr2(t)ωmδqγδp+G(δa+δa)+ξ,δ˙a=(κ+iΔc)δa+iGδq+2κain,
u˙(t)=A(t)u(t)+N(t),
A(t)=(κΔc00Δcκ2G0000ωm2G0ωr2(t)/ωmγ).
u(t)=L(t)u(0)+L(t)0tdτL1(τ)N(τ),
Rij(t)=[ui(t)uj(t)+uj(t)ui(t)]/2.
R=L(t)R(0)LT(t)+L(t)M(t)LT(t),
M(t)=12[W(t)+WT(t)],W(t)=0tdτ0tdτL1(τ)C(τ,τ)[L1(τ)]T.
Nij(τ)Nij(τ)+NijT(τ)NijT(τ)/2=Dijδ(ττ).
M(t)=0tdτL1(τ)D(τ)[L1(τ)]T.
R˙(t)=A(t)R(t)+R(t)AT(t)+D.
ΔX2=cos2θδq2+sin2θδp2+12sin2θ(δqδp+δpδq)=cos2θR33+sin2θR44+12sin2θ(R34+R43).
V˙(t)=cos2θR˙33+sin2θR˙44+12sin2θ(R˙34+R˙43),
R˙33=ωm(R34+R43),R˙44=2γR44ωr(t)2ωm(R34+R43)+2G(R14+R41)+γ(2n¯mT+1),R˙34=ωmR44ωr(t)2ωmR33γR34+2GR31,R˙43=ωmR44ωr(t)2ωmR33γR43+2GR13.
V˙(t)=2γR44ωr2(t)ωm(R34+R43)+2G(R14+R41)+γ(2n¯mT+1).
V˙(t)2γR44c|R34+R43|2,
ωr2(t)=cωm(R34+R43).
S=10log10(Δp2/Δpzp2).
Ht=H+ωsasasgsasasq+iEp(eiωptash.c.),
δ˙as=(κs+iΔs)δas+iGsδq+2κsas,in,
δ˙as=(κs+iΔs)δas+iGs2δb+2κsas,in,
δas=iGs2(κs+iΔs)δb+2κsκs+iΔsas,in.
δas,out=iκsGsκs+iΔsδb+(2κsκs+iΔs1)as,in.
δXs,out(ϕ)=Gsκs(Δscosϕ+κssinϕ)Δs2+κs2δq+Gsκs(Δssinϕκscosϕ)Δs2+κs2δp+Fad,
δXs,out(ϕ)=Gsκs(Δssinϕκscosϕ)Δs2+κs2δp+Fad.
ncd=i(|βl|eiϕlδas,out|βl|eiϕlδas,out),
(Δncd)2=4|βl|2ΔXs,out2(ϕl+π/2),
R˙(t)=A(t)R(t)+R(t)AT(t)+D,
A(t)=(kΔc0000Δcκ2G000000ωm002G0ωr2(t)ωmγ2Gs00000κsΔs002Gs0Δsκs).
H=Δcδaδa+Δdδdδd+ωm2(δp2+δq2)G(δa+δa)δqGd(δd+δd)δq,
δ˙q=ωmδp,δ˙p=ωmδqγδp+G(δa+δa)+Gd(δd+δd)+ξ,δ˙d=(κd+iΔd)δd+iGdδq+2κdin.
δ˙q=ωmδp,δ˙p=(ωm2Gd2ΔdΔd2+κd2)δqγδp+G(δa+δa)+ξ.
H=Δδaδa+ωm2δp2+ωr2(t)2ωmδq2G(δa+δa)δq,