Abstract

The interaction of a single-mode field with both a weak Kerr medium and a parametric nonlinearity in an intrinsically nonlinear optomechanical system is studied. The nonlinearities due to the optomechanical coupling and Kerr-down conversion lead to bistability and tristability in the mean intracavity photon number. Also, our work demonstrates that the lower bound of the resolved sideband regime and the minimum attainable phonon number can be less than those of a bare cavity by controlling the parametric nonlinearity and the phase of the driving field. Moreover, we find that in the system under consideration the degree of entanglement between the mechanical and optical modes is dependent on the two stability parameters of the system. For both cooling and entanglement, while parametric nonlinearity increases the optomechanical coupling, the weak Kerr nonlinearity is very useful for extending the domain of the stability region to the desired range in which the minimum effective temperature and maximal entanglement are attainable. Also, as shown in this paper, the present scheme allows us to have significant entanglement in the tristable regime for the lower and middle branches, which makes the current scheme distinct from the bare optomechanical system.

© 2014 Optical Society of America

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2012 (2)

A. Xuereb, M. Barbier, and M. Paternostro, “Multipartite optomechanical entanglement from competing nonlinearities,” Phys. Rev. A 86, 013809 (2012).
[CrossRef]

J. Sheng, U. Khadka, and M. Xiao, “Realization of all-optical multistate switching in an atomic coherent medium,” Phys. Rev. Lett. 109, 223906 (2012).
[CrossRef]

2011 (6)

J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring Resonators,” Opt. Express 19, 11415–11421 (2011).
[CrossRef]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[CrossRef]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[CrossRef]

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[CrossRef]

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[CrossRef]

2010 (2)

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

T. Kumar, A. Bhattacherjee, and M. Mohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010).
[CrossRef]

2009 (5)

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[CrossRef]

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Z. R. Gong, H. Ian, Yu.-x. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[CrossRef]

A. Mari and J. Eisert, “Gently modulating optomechanical system,” Phys. Rev. Lett. 103, 213603 (2009).
[CrossRef]

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Quantum effects in optomechanical systems,” Adv. At. Mol. Opt. Phys. 57, 33–86 (2009).
[CrossRef]

2008 (4)

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
[CrossRef]

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101, 263602 (2008).
[CrossRef]

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

2007 (4)

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

F. Marquardt, J. P. Chen, A. A. Clerck, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[CrossRef]

2006 (5)

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef]

D. Kleckner, W. Marshall, M. J. A. de Dood, K. N. Dinyari, B.-J. Pors, W. T. M. Irvine, and D. Bouwmeester, “High finesse opto-mechanical cavity with a movable thirty-micron-size mirror,” Phys. Rev. Lett. 96, 173901 (2006).
[CrossRef]

F. Marquardt, J. G. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[CrossRef]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

2005 (1)

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef]

2004 (3)

C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432, 1002–1005 (2004).
[CrossRef]

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[CrossRef]

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).
[CrossRef]

2002 (1)

V. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in power recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
[CrossRef]

2001 (1)

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

1997 (2)

S. Mancini, V. I. Manko, and P. Tombesi, “Ponderomotive control of quantum macroscopic coherence,” Phys. Rev. A 55, 3042 (1997).
[CrossRef]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
[CrossRef]

1996 (1)

W. Leon’ski, “Fock states in a Kerr medium with parametric pumping,” Phys. Rev. A 54, 3369–3372 (1996).
[CrossRef]

1995 (1)

C. K. Law, “Interaction between a moving mirror and radiation pressurer: a Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[CrossRef]

1994 (2)

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[CrossRef]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[CrossRef]

1993 (2)

T. Kaino and S. Tomaru, “Organic material for nonlinear optics,” Adv. Mater. 5, 172–178 (1993).
[CrossRef]

B. Wielinga and G. J. Milburn, “Quantum tunneling in a Kerr medium with parametric pumping,” Phys. Rev. A 48, 2494–2496 (1993).
[CrossRef]

1989 (1)

P. D. Townsend, G. L. Baker, J. L. Jackel, J. A. Shelburne, and S. Etemad, “Nonlinear optical properties of organic materials II,” Proc. SPIE 1147, 256 (1989).
[CrossRef]

1985 (1)

1983 (1)

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550–1553 (1983).
[CrossRef]

Adesso, G.

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).
[CrossRef]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[CrossRef]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[CrossRef]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

Ansmann, M.

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Arcizet, O.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Aspelmeyer, M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[CrossRef]

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[CrossRef]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Bahrampour, A. R.

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[CrossRef]

Baker, G. L.

P. D. Townsend, G. L. Baker, J. L. Jackel, J. A. Shelburne, and S. Etemad, “Nonlinear optical properties of organic materials II,” Proc. SPIE 1147, 256 (1989).
[CrossRef]

Barbarino, S.

Barbier, M.

A. Xuereb, M. Barbier, and M. Paternostro, “Multipartite optomechanical entanglement from competing nonlinearities,” Phys. Rev. A 86, 013809 (2012).
[CrossRef]

Bauerle, D.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Bhattacherjee, A.

T. Kumar, A. Bhattacherjee, and M. Mohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010).
[CrossRef]

Bialczak, R. C.

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Blaser, F.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Bohm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Borkje, K.

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[CrossRef]

Bose, S.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
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A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Ortlieb, A.

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[CrossRef]

Painter, O.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[CrossRef]

Paternostro, M.

A. Xuereb, M. Barbier, and M. Paternostro, “Multipartite optomechanical entanglement from competing nonlinearities,” Phys. Rev. A 86, 013809 (2012).
[CrossRef]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[CrossRef]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Pinard, M.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[CrossRef]

Pors, B.-J.

D. Kleckner, W. Marshall, M. J. A. de Dood, K. N. Dinyari, B.-J. Pors, W. T. M. Irvine, and D. Bouwmeester, “High finesse opto-mechanical cavity with a movable thirty-micron-size mirror,” Phys. Rev. Lett. 96, 173901 (2006).
[CrossRef]

Rabl, P.

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[CrossRef]

Reynaud, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[CrossRef]

Rokhsari, H.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef]

Rugar, D.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Safavi-Naeini, A. H.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[CrossRef]

Sank, D.

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Schwab, K. C.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Serafini, A.

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).
[CrossRef]

Shelburne, J. A.

P. D. Townsend, G. L. Baker, J. L. Jackel, J. A. Shelburne, and S. Etemad, “Nonlinear optical properties of organic materials II,” Proc. SPIE 1147, 256 (1989).
[CrossRef]

Sheng, J.

J. Sheng, U. Khadka, and M. Xiao, “Realization of all-optical multistate switching in an atomic coherent medium,” Phys. Rev. Lett. 109, 223906 (2012).
[CrossRef]

Simmonds, R. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

Simon, C.

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[CrossRef]

Sirois, A. J.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

Strigin, S. E.

V. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in power recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
[CrossRef]

Sun, C. P.

Z. R. Gong, H. Ian, Yu.-x. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[CrossRef]

Teufel, J. D.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

Tomaru, S.

T. Kaino and S. Tomaru, “Organic material for nonlinear optics,” Adv. Mater. 5, 172–178 (1993).
[CrossRef]

Tombesi, P.

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Quantum effects in optomechanical systems,” Adv. At. Mol. Opt. Phys. 57, 33–86 (2009).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
[CrossRef]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

S. Mancini, V. I. Manko, and P. Tombesi, “Ponderomotive control of quantum macroscopic coherence,” Phys. Rev. A 55, 3042 (1997).
[CrossRef]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[CrossRef]

Townsend, P. D.

P. D. Townsend, G. L. Baker, J. L. Jackel, J. A. Shelburne, and S. Etemad, “Nonlinear optical properties of organic materials II,” Proc. SPIE 1147, 256 (1989).
[CrossRef]

Vahala, K. J.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef]

Vanner, M. R.

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Vedral, V.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

Vignes, E.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550–1553 (1983).
[CrossRef]

Vitali, D.

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Quantum effects in optomechanical systems,” Adv. At. Mol. Opt. Phys. 57, 33–86 (2009).
[CrossRef]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[CrossRef]

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

Vyatchanin, S. P.

V. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in power recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
[CrossRef]

Walther, H.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550–1553 (1983).
[CrossRef]

Wang, H.

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Weides, M.

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Wenner, J.

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Whittaker, J. D.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

Wielinga, B.

B. Wielinga and G. J. Milburn, “Quantum tunneling in a Kerr medium with parametric pumping,” Phys. Rev. A 48, 2494–2496 (1993).
[CrossRef]

Wilson-Rae, I.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101, 263602 (2008).
[CrossRef]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

Xiao, M.

J. Sheng, U. Khadka, and M. Xiao, “Realization of all-optical multistate switching in an atomic coherent medium,” Phys. Rev. Lett. 109, 223906 (2012).
[CrossRef]

Xuereb, A.

A. Xuereb, M. Barbier, and M. Paternostro, “Multipartite optomechanical entanglement from competing nonlinearities,” Phys. Rev. A 86, 013809 (2012).
[CrossRef]

Yang, L.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef]

Zeilinger, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Zoller, P.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 1991).

Zwerger, W.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

Adv. At. Mol. Opt. Phys. (1)

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Quantum effects in optomechanical systems,” Adv. At. Mol. Opt. Phys. 57, 33–86 (2009).
[CrossRef]

Adv. Mater. (1)

T. Kaino and S. Tomaru, “Organic material for nonlinear optics,” Adv. Mater. 5, 172–178 (1993).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (9)

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[CrossRef]

A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[CrossRef]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432, 1002–1005 (2004).
[CrossRef]

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

V. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in power recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
[CrossRef]

Phys. Rev. A (16)

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).
[CrossRef]

C. K. Law, “Interaction between a moving mirror and radiation pressurer: a Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[CrossRef]

S. Mancini, V. I. Manko, and P. Tombesi, “Ponderomotive control of quantum macroscopic coherence,” Phys. Rev. A 55, 3042 (1997).
[CrossRef]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
[CrossRef]

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[CrossRef]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[CrossRef]

A. Xuereb, M. Barbier, and M. Paternostro, “Multipartite optomechanical entanglement from competing nonlinearities,” Phys. Rev. A 86, 013809 (2012).
[CrossRef]

T. Kumar, A. Bhattacherjee, and M. Mohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010).
[CrossRef]

B. Wielinga and G. J. Milburn, “Quantum tunneling in a Kerr medium with parametric pumping,” Phys. Rev. A 48, 2494–2496 (1993).
[CrossRef]

W. Leon’ski, “Fock states in a Kerr medium with parametric pumping,” Phys. Rev. A 54, 3369–3372 (1996).
[CrossRef]

Z. R. Gong, H. Ian, Yu.-x. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[CrossRef]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[CrossRef]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[CrossRef]

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
[CrossRef]

Phys. Rev. Lett. (14)

A. Mari and J. Eisert, “Gently modulating optomechanical system,” Phys. Rev. Lett. 103, 213603 (2009).
[CrossRef]

F. Marquardt, J. P. Chen, A. A. Clerck, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[CrossRef]

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101, 263602 (2008).
[CrossRef]

J. Sheng, U. Khadka, and M. Xiao, “Realization of all-optical multistate switching in an atomic coherent medium,” Phys. Rev. Lett. 109, 223906 (2012).
[CrossRef]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

F. Marquardt, J. G. Harris, and S. M. Girvin, “Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities,” Phys. Rev. Lett. 96, 103901 (2006).
[CrossRef]

C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, “Self-induced oscillations in an optomechanical system driven by bolometric backaction,” Phys. Rev. Lett. 101, 133903 (2008).
[CrossRef]

D. Kleckner, W. Marshall, M. J. A. de Dood, K. N. Dinyari, B.-J. Pors, W. T. M. Irvine, and D. Bouwmeester, “High finesse opto-mechanical cavity with a movable thirty-micron-size mirror,” Phys. Rev. Lett. 96, 173901 (2006).
[CrossRef]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[CrossRef]

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[CrossRef]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550–1553 (1983).
[CrossRef]

Proc. SPIE (1)

P. D. Townsend, G. L. Baker, J. L. Jackel, J. A. Shelburne, and S. Etemad, “Nonlinear optical properties of organic materials II,” Proc. SPIE 1147, 256 (1989).
[CrossRef]

Other (4)

L. Landau and E. Lifshitz, Statistical Physics (Pergamon, 1958).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

A. Hurwitz, Selected Papers on Mathematical Trends in Control Theory, R. Bellman and R. Kabala, eds. (Dover, 1964).

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 1991).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic picture of the setup studied in the text. The cavity contains a Kerr-down conversion system that is pumped by a coupling field to produce parametric oscillation and induce Kerr nonlinearity in the cavity.

Fig. 2.
Fig. 2.

Mean intracavity photon number as a function of normalized bare detuning Δ0/ωm: (a) for different values of the anharmonicity parameter χ with G=0.6κ and θ=π/2, (b) for different values of the parametric nonlinearity G with χ=0.1Hz and θ=π/2, and (c) for different values of θ with G=1.1κ and χ=0.04Hz.

Fig. 3.
Fig. 3.

Mean photon number Ia versus the input power P at Δ0=2.5ωm. The solid and dashed lines correspond to the stable and unstable branches, respectively. The parameters are G=κ, χ=0.05Hz, and κ=0.9ωm. Other parameters are the same as those in Fig. 2.

Fig. 4.
Fig. 4.

Plot of the effective temperature T versus Δ/ωm for different values of G. The red dashed curve corresponds to the bare cavity. The parameters are : χ=0.05Hz, θ=0.81π for G=0.6κ and θ=0.71π for G=0.8κ, T0=400mK, P=5mW, and κ=0.3ωm. Other parameters are the same as those in Fig. 3.

Fig. 5.
Fig. 5.

Plot of the effective temperature T versus Δ for environment temperature (a) T0=400(mK) and (b) T0=25(mK) for different values of χ. The red dashed curve corresponds to the cavity containing only the gain nonlinearity. The parameters are G=0.8κ, θ=3π/4, and P=5mW. Other parameters are the same as those in Fig. 3, and the stability parameter is fixed to be η1=0.99.

Fig. 6.
Fig. 6.

Plot of the stability parameter η1 versus Δ/ωm and input power P for χ=0.03. Other parameters are the same as those in Fig. 5(a).

Fig. 7.
Fig. 7.

Plot of (a) effective temperature T and (b) logarithmic negativity versus the input power P for Δ=0.5ωm, G=1.3κ, θ=0.67π, and χ=0.05. Other parameters are the same as those in Fig. 4.

Fig. 8.
Fig. 8.

Plot of the logarithmic negativity versus the normalized effective detuning Δ/ωm for different values of θ. The parameters are P=3mW, G=1.3κ, and χ=0.05. Other parameters are the same as those in Fig. 4.

Fig. 9.
Fig. 9.

Contour plot of the logarithmic negativity versus the normalized effective detuning Δ/ωm and input power P for (a) G=0.6κ and (b) G=κ. The parameters are θ=0.67π, χ=0.05. Other parameters are the same as those in Fig. 4.

Fig. 10.
Fig. 10.

Logarithmic negativity versus the normalized effective detuning Δ/ωm for χ=0, χ=0.03, and χ=0.05. The parameters are θ=0.67π, G=κ, and P=6mW. Other parameters are the same as those in Fig. 4.

Fig. 11.
Fig. 11.

(a) Logarithmic negativity and (b) stability parameter η1 as functions of the input power P corresponding to the three stable branches for θ=0.57π in Fig. 3. The dashed line corresponds to the unstable region.

Tables (1)

Tables Icon

Table 1. Calculated Logarithmic Negativities, Normalized Effective Detunings, Normalized Optomechanical Couplings, and the Two Stability Parameters for the Input Powers P=2.5mW and P=5mW in Fig. 9

Equations (29)

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H=H0+H1,
H0=(ω0ωL)aa+ωm2(q2+p2)g0aaq+iε(aa),
H1=iG(eiθa2eiθa2)+χa2a2.
q˙=ωmp,
p˙=ωmq+g0aaγmp+ξ,
a˙=i(ω0ωL)a+ig0qa+ε2iχaa2+2Gaeiθκa+2κain,
ain(t)ain(t)=δ(tt),
ain(t)ain(t)=ain(t)ain(t)=0.
ξ(t)ξ(t)=γmπωmωeiω(tt)[coth(ω2kBT0)+1]dω,
ps=0,qs=g0ωmas2,
as=κiΔ+2GeiθΔ2+κ24G2ε,
I±=(4Gsinθ2Δ0)±(2GsinθΔ0)23κ23(2χg02/ωm).
a=as+δa,q=qs+δq,p=ps+δp.
u˙=Mu(t)+n(t),
M=(0ωm00ωmγmg1000κ+ΓrΔ1+Γig10Δ1+ΓiκΓr),
s1=κ2+Δ12|Γ|2>0,
s2=ωm(κ2+Δ12|Γ|2)g02(Δ1+Γi)>0,
s3=2κγm{(s1ωm2)2+(γm+2κ)(γms1+2κωm2)}+g02(Δ1+Γi)ωm(2κ+γm)2>0.
η1=1g12(Δ1+Γi)ωm(κ2+Δ12|Γ|2),
η2==1|Γ|2κ2+Δ12.
MV+VMT=D,
ωcoth(ω2kBT0)ωm2kBT0ωmωm(2n¯+1),
U=ωm2(δq2+δp2)=ωm(neff+12),
EN=max[0,ln2η],
(VAVCVCTVB),
V11=η1ωm2+(Δ1+Γi)2+(κ+Γr)24η1ωm(Δ1+Γi),
V22=ωm2+(Δ1+Γi)2+(κ+Γr)24ωm(Δ1+Γi),
neff=(Δ1+Γiωm)2+κ+24ωm(Δ1+Γi),
nmin=12(ωm2+κ+2ωm1),

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