Skip to main content
SpringerLink
Log in

Normal Mode Splitting in a Cavity Optomechanical System with a Cubic Anharmonic Oscillator

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

The strong coupling between a macroscopic mechanical oscillator and a cavity field is essential for many quantum phenomena in a cavity optomechanical system. In this work, we discuss the normal mode splitting in a cavity optomechanical system with a cubic nonlinear movable mirror. We study how the mechanical nonlinearity affects the normal-mode splitting behavior of the movable mirror and the output field. We find that the mechanical nonlinearity can increase the peak separation in the spectra of the movable mirror and the output field. We also find that the heights and linewidths of the two peaks are very sensitive to the mechanical nonlinearity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Aspelmeyer, M., Kippenberg, T.J., Marquardt, F.: Cavity optomechanics. Rev. Mod. Phys. 86, 1391 (2014)

    Article  ADS  Google Scholar 

  2. Clark, J.B., Lecocq, F., Simmonds, R.W., Aumentado, J., Teufel, J.D.: Sideband cooling beyond the quantum backaction limit with squeezed light. Nature (London) 541, 191–195 (2017)

    Article  ADS  Google Scholar 

  3. Qiu, L., Shomroni, I., Seidler, P., Kippenberg, T.J.: Laser cooling of a nanomechanical oscillator to its zero-point energy. Phys. Rev. Lett. 124, 173601 (2020)

    Article  ADS  Google Scholar 

  4. Wollman, E.E., Lei, C.U., Weinstein, A.J., Suh, J., Kronwald, A., Marquardt, F., Clerk, A.A., Schwab, K.C.: Quantum squeezing of motion in a mechanical resonator. Science 349, 952 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Ockeloen-Korppi, C.F., Damskägg, E., Pirkkalainen, J.M., Asjad, M., Clerk, A.A., Massel, F., Woolley, M.J., Sillanpää, M. A.: Stabilized entanglement of massive mechanical oscillators. Nature (London) 556, 478 (2018)

    Article  ADS  Google Scholar 

  6. Safavi-Naeini, A.H., Gröblacher, S., Hill, J.T., Chan, J., Aspelmeyer, M., Painter, O.: Squeezed light from a silicon micromechanical resonator. Nature (London) 500, 185 (2013)

    Article  ADS  Google Scholar 

  7. Marquardt, F., Chen, J.P., Clerk, A.A., Girvin, S.M.: Quantum theory of cavity-assisted sideband cooling of mechanical motion. Phys. Rev. Lett. 99, 093902 (2007)

    Article  ADS  Google Scholar 

  8. Dobrindt, J.M., Wilson-Rae, I., Kippenberg, T.J.: Parametric normal-mode splitting in cavity optomechanics. Phys. Rev. Lett. 101, 263602 (2008)

    Article  ADS  Google Scholar 

  9. Huang, S., Agarwal, G.S.: Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity. Phys. Rev. A 80, 033807 (2009)

    Article  ADS  Google Scholar 

  10. Bhattacherjee, A.B.: Cavity quantum optomechanics of ultracold atoms in an optical lattice: Normal-mode splitting. Phys. Rev. A 80, 043607 (2009)

    Article  ADS  Google Scholar 

  11. Kumar, T., Bhattacherjee, A.B.: ManMohan: Dynamics of a movable micromirror in a nonlinear optical cavity. Phys. Rev. A 81, 013835 (2010)

    Article  ADS  Google Scholar 

  12. Liu, Y.C., Xiao, Y.F., Luan, X.S., Gong, Q.H., Wong, C.W.: Coupled cavities for motional ground-state cooling and strong optomechanical coupling. Phys. Rev. A 91, 033818 (2015)

    Article  ADS  Google Scholar 

  13. Han, Y., Cheng, J., Zhou, L.: Normal-mode splitting in the atom-assisted optomechanical cavity. Phys. Scr. 88, 065401 (2013)

    Article  ADS  Google Scholar 

  14. Rossi, M., Kralj, N., Zippilli, S., Natali, R., Borrielli, A., Pandraud, G., Serra, E., Giuseppe, G.D., Vitali, D.: Normal-mode splitting in a weakly coupled optomechanical system. Phys. Rev. Lett. 120, 073601 (2018)

    Article  ADS  Google Scholar 

  15. Huang, S., Chen, A.: Cooling of a mechanical oscillator and normal mode splitting in optomechanical systems with coherent feedback. Appl. Sci. 9, 3402 (2019)

    Article  Google Scholar 

  16. Zhang, Z.C., Wang, Y.P., Yu, Y.F., Zhang, Z.M.: Normal-mode splitting in a weakly coupled electromechanical system with a mechanical modulation. Ann. Phys. 531, 1800461 (2019)

    Article  MathSciNet  Google Scholar 

  17. Ullah, K.: The occurrence of multistability and normal mode splitting in an optomechanical system. Phys. Lett. A 383, 3074–3079 (2019)

    Article  ADS  Google Scholar 

  18. Gröblacher, S., Hammerer, K., Vanner, M.R., Aspelmeyer, M.: Observation of strong coupling between a micromechanical resonator and an optical cavity field. Nature (London) 460, 724–727 (2009)

    Article  ADS  Google Scholar 

  19. Verhagen, E., Deléglise, S., Weis, S., Schliesser, A., Kippenberg, T.J.: Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode. Nature (London) 482, 63–67 (2012)

    Article  ADS  Google Scholar 

  20. Sommer, A.R., Meyer, N., Quidant, R.: Strong optomechanical coupling at room temperature by coherent scattering. Nat. Commun. 12, 276 (2021)

    Article  Google Scholar 

  21. Enzian, G., Szczykulska, M., Silver, J., Del Bino, L., Zhang, S., Walmsley, I.A., Del’Haye, P., Vanner, M.R.: Observation of brillouin optomechanical strong coupling with an 11 GHz mechanical mode. Optica 6, 7–14 (2019)

    Article  ADS  Google Scholar 

  22. Teufel, J.D., Li, D., Allman, M.S., Cicak, K., Sirois, A.J., Whittaker, J.D., Simmonds, R.W.: Circuit cavity electromechanics in the strong-coupling regime. Nature (London) 471, 204 (2011)

    Article  ADS  Google Scholar 

  23. Sanchez-Mondragon, J.J., Narozhny, N.B., Eberly, J.H.: Theory of spontaneous-emission line shape in an ideal cavity. Phys. Rev. Lett. 51, 550 (1983)

    Article  ADS  Google Scholar 

  24. Agarwal, G.S.: Vacuum-field rabi splittings in microwave absorption by Rydberg atoms in a cavity. Phys. Rev. Lett. 53, 1732 (1984)

    Article  ADS  Google Scholar 

  25. Thompson, R.J., Rempe, G., Kimble, H.J.: Observation of normal-mode splitting for an atom in an optical cavity. Phys. Rev. Lett. 68, 1132 (1992)

    Article  ADS  Google Scholar 

  26. Reithmaier, J.P., Sȩk, G., Löffler1, A., Hofmann, C., Kuhn, S., Reitzenstein, S., Keldysh, L.V., Kulakovskii, V.D., Reinecke, T.L., Forchel, A.: Strong coupling in a single quantum dot-semiconductor microcavity system. Nature (London) 432, 197–200 (2004)

  27. Yoshie, T., Scherer, A., Hendrickson, J., Khitrova, G., Gibbs, H.M., Rupper, G., Ell, C., Shchekin, O.B., Deppe, D.G.: Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity. Nature (London) 432, 200–203 (2004)

    Article  ADS  Google Scholar 

  28. Jacobs, K., Landahl, A.J.: Engineering giant nonlinearities in quantum nanosystems. Phys. Rev. Lett. 103, 067201 (2009)

    Article  ADS  Google Scholar 

  29. Rips, S., Wilson-Rae, I., Hartmann, M.J.: Nonlinear nanomechanical resonators for quantum optoelectromechanics. Phys. Rev. A 89, 013854 (2014)

    Article  ADS  Google Scholar 

  30. Gieseler, J., Novotny, L., Quidant, R.: Thermal nonlinearities in a nanomechanical oscillator. Nat. Phys. 9, 806 (2013)

    Article  Google Scholar 

  31. Latmiral, L., Armata, F., Genoni, M.G., Pikovski, I., Kim, M.S.: Probing anharmonicity of a quantum oscillator in an optomechanical cavity. Phys. Rev. A 93, 052306 (2016)

    Article  ADS  Google Scholar 

  32. Huang, S., Hao, H., Chen, A.: The optomechanical response of a cubic anharmonic oscillator. Appl. Sci. 5719, 10 (2020)

    Google Scholar 

  33. Grimm, M., Bruder, C., Lörch, N.: Optomechanical self-oscillations in an anharmonic potential: engineering a nonclassical steady state. J. Opt. 18, 094004 (2016)

    Article  ADS  Google Scholar 

  34. Rips, S., Hartmann, M.J.: Quantum information processing with nanomechanical qubits. Phys. Rev. Lett. 110, 120503 (2013)

    Article  ADS  Google Scholar 

  35. Rips, S., Kiffner, M., Wilson-Rae, I., Hartmann, M.J.: Steady-state negative Wigner functions of nonlinear nanomechanical oscillators. New. J. Phys. 14, 023042 (2012)

    Google Scholar 

  36. Lü, X. Y., Liao, J.Q., Tian, L., Nori, F.: Steady-state mechanical squeezing in an optomechanical system via Duffing nonlinearity. Phys. Rev. A 91, 013834 (2015)

    Article  ADS  Google Scholar 

  37. Alvarez, G.: Coupling-constant behavior of the resonances of the cubic anharmonic oscillator. Phys. Rev. A 37, 4079 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  38. Wartak, M.S., Krzeminski, S.: On tunnelling in the cubic potential. J. Phys. Math. Gen. 22, L1005 (1989)

    Article  ADS  Google Scholar 

  39. Alvarez, G.: Bender-Wu branch points in the cubic oscillator. J. Phys. A: Math. Gen. 28, 4589 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Mera, H., Pedersen, T.G., Nikolić, B. K.: Nonperturbative quantum physics from low-order perturbation theory. Phys. Rev. Lett. 115, 143001 (2015)

    Article  ADS  Google Scholar 

  42. Cveticanin, L., Zukovic, M., Mester, G., Biro, I., Sarosi, J.: Oscillators with symmetric and asymmetric quadratic nonlinearity. Acta Mech. 227, 1727–1742 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pal, A., Bhattacharjee, J.K.: Quantum dynamics in a cubic potential in the semi-classical limit: Smearing of the homoclinic bifurcation. Phys. Open 6, 100047 (2021)

    Article  Google Scholar 

  44. Law, C.K.: Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium. Phys. Rev. A 49, 433 (1994)

    Article  ADS  Google Scholar 

  45. Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 51, 793 (1948)

    MATH  Google Scholar 

  46. Giovannetti, V., Vitali, D.: Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion. Phys. Rev. A 63, 023812 (2001)

    Article  ADS  Google Scholar 

  47. 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)

    Article  ADS  MathSciNet  Google Scholar 

  48. Walls, D.F., Milburn, G.J.: Quantum Optics, pp 121–124. Springer, Berlin Germany (1998)

    MATH  Google Scholar 

  49. Zhou, X., Hocke, F., Schliesser, A., Marx, A., Huebl, H., Gross, R., Kippenberg, T.J.: Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics. Nat. Phys. 9, 179 (2013)

    Article  Google Scholar 

  50. Børkje, K.: Critical quantum fluctuations and photon antibunching in optomechanical systems with large single-photon cooperativity. Phys. Rev. A 101, 053833 (2020)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This study was funded by Zhejiang Provincial Natural Science Foundation of China [Grant numbers LY21A040007, LZ20A040002], by Science Foundation of Zhejiang Sci-Tech University [Grant numbers 18062121-Y, 17062071-Y], and by the National Natural Science Foundation of China [Grant numbers 11775190, 91636108].

Author information

Authors and Affiliations

  1. Department of Physics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Hongmiao Hao, Sumei Huang & Aixi Chen

Authors
  1. Hongmiao Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sumei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aixi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sumei Huang.

Ethics declarations

Conflict of Interests

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hao, H., Huang, S. & Chen, A. Normal Mode Splitting in a Cavity Optomechanical System with a Cubic Anharmonic Oscillator. Int J Theor Phys 60, 2766–2777 (2021). https://doi.org/10.1007/s10773-021-04855-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-021-04855-4

Keywords

Search

Navigation