Abstract

We present a scheme for the electromagnetically induced transparency (EIT)-like nonlinear ground-state cooling in a double-cavity optomechanical system in which an optical cavity mode is coupled parametrically to the square of the position of a mechanical oscillator, an additional auxiliary cavity is coupled to the optomechanical cavity. The optimum cooling conditions is derived, based on which the heating process can be well suppressed and the mechanical resonator can be cooled with an optimal effect to near its ground state through EIT-like cooling mechanism even in unresolved sideband regime. It is demonstrated by numerical simulations that not only the average phonon number of steady state is lower than that of single-cavity optomechanical system, but also the cooling rate is greatly faster than that of the linear optomechanical coupling due to the two-phonon cooling process in the quadratic coupling. Also, the ground-state cooling is achievable even with a relatively weak quadratic coupling strengthby tunning the coupling between two cavities to reach the optimum cooling conditions, thus provides an solution for overcoming the limitations of weak quadratic coupling rate in experiments. The proposed approach provides a platform for quantum manipulation of macroscopic mechanical devices beyond the resolved sideband limit and linear coupling regime.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53 (2018).
[Crossref] [PubMed]

2017 (2)

J. B. Clark, F. Lecocq, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Sideband cooling beyond the quantum backaction limit with squeezed light,” Nature 541, 191 (2017).
[Crossref] [PubMed]

J. S. Feng, L. Tan, H. Q. Gu, and W. M. Liu, “Auxiliary-cavity-assisted ground-state cooling of an optically levitated nanosphere in the unresolved-sideband regime,” Phys. Rev. A 96, 063818 (2017).
[Crossref]

2016 (1)

R. W. Peterson, T. P. Purdy, N. S. Kampel, R.W. Andrews, P. L. Yu, K. W. Lehnert, and C.A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

2015 (6)

Y. C. Liu, Y. F. Xiao, X. S. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 050305 (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]

Y. Yan, W. J. Gu, and G. X. Li, “Entanglement transfer from two-mode squeezed vacuum light to spatially separated mechanical oscillators via dissipative optomechanical coupling,” Sci. China Phys. Mech. Astron. 58, 050306 (2015).
[Crossref]

X. 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] [PubMed]

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

W. J. Gu, Z. Yi, L. H. Sun, and D. H. Xu, “Mechanical cooling in single-photon optomechanics with quadratic nonlinearity,” Phys. Rev. A 92, 023811 (2015).
[Crossref]

2014 (6)

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

S. Zhang, J. Q. Zhang, J. Zhang, C. W. Wu, W. Wu, and P. X. Chen, “Ground state cooling of an optomechanical resonator assisted by a Λ-type atom,” Opt. Express 22, 28118 (2014).
[Crossref] [PubMed]

Y. Guo, K. Li, W. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (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]

P. Sekatski, M. Aspelmeyer, and N. Sangouard, “Macroscopic optomechanics from displaced single-photon entanglement,” Phys. Rev. Lett. 112, 080502 (2014).
[Crossref]

T. Ojanen and K. Børkje, “Ground state cooling of mechanical motion in the unresolved sideband regime by use of optomechanically induced transparency,” Phys. Rev. A 90, 013824 (2014).
[Crossref]

2013 (5)

W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87, 025804 (2013).
[Crossref]

T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
[Crossref]

M. Y. Yan, H. K. Li, Y. C. Liu, W. L. Jin, and Y. F. Xiao, “Dissipative optomechanical coupling between a single-wall carbon nanotube and a high-Q microcavity,” Phys. Rev. A 88, 023802 (2013).
[Crossref]

J. J. Li and K. D. Li, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223 (2013).
[Crossref]

H. K. Li, X. X. Ren, Y. C. Liu, and Y. F. Xiao, “Effective photon-photon interactions in largely detuned optomechanics,” Phys. Rev. A 88, 053850 (2013).
[Crossref]

2012 (6)

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65, 29 (2012).
[Crossref]

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

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “A reversible optical to microwave quantum interface,” Phys. Rev. Lett. 109, 130503 (2012).
[Crossref]

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6, 768 (2012).
[Crossref]

B. Pepper, R. Ghobadi, E. Jeffrey, C. Simon, and D. Bouwmeester, “Optomechanical superpositions via nested interferometry,” Phys. Rev. Lett. 109, 023601 (2012).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon regime of optomechanics,” Phys. Rev. A 85, 051803 (2012).
[Crossref]

2011 (7)

A. Xuereb, R. Schnabel, and K. Hammerer, “Dissipative optomechanics in a Michelson-Sagnac interferometer,” Phys. Rev. Lett. 107, 213604 (2011).
[Crossref] [PubMed]

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

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects, optomechanical superpositions via nested interferometry,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing an optical pulse as a mechanical excitation in a silica pptomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[Crossref]

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, and G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107, 023601 (2011).
[Crossref] [PubMed]

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” Physics 15, 205 (2011).

2010 (7)

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 104, 213603 (2010).
[Crossref]

A. A. Clerk, S. M. Girvin, and F. Marquardt, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
[Crossref]

K. Hammerer, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, and T. W. Hansch, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803 (2010).
[Crossref]

T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z. Y. Ma, and D. M. Stamper-Kurn, “Tunable cavity optomechanics with ultracold atoms,” Phys. Rev. Lett. 105, 133602 (2010).
[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 (London) 464, 697 (2010).
[Crossref]

2009 (5)

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[Crossref]

C. Genes, H. Ritsch, and D. Vitali, “Micromechanical oscillator ground-state cooling via resonant intracavity optical gain or absorption,” Phys. Rev. A 80, 061803 (2009).
[Crossref]

F. Helmer, M. Mariantoni, E. Solano, and F. Marquardt, “Quantum nondemolition photon detection in circuit QED and the quantum Zeno effect,” Phys. Rev. A 79, 052115 (2009).
[Crossref]

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum Limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
[Crossref] [PubMed]

2008 (6)

J. D. Thompson, B. M. Zwickl, and A. M. Jayich, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature (London) 452, 72 (2008).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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K. W. Murch, K. L. Moore, and S. Gupta, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561 (2008).
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J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin, and J. G. E. HarrisStrong, “Strong dispersive coupling of a high finesse cavity to a micromechanical membrane,” Nature (London) 452, 72 (2008).
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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).
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2007 (4)

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
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K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58 (7), 36 (2005).
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2004 (1)

M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
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1998 (1)

S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80, 688 (1998).
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V. V. Dodonov and S. S. Mizrahi, “Exact stationary photon distributions due to competition between one- and two-photon absorption and emission,” J. Phys. A 30, 5657 (1997).
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1994 (1)

R. L. de Matos Filho and W. Vogel, “Second-sideband laser cooling and nonclassical motion of trapped ions,” Phys. Rev. A 50, R1988 (1994).
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Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “A reversible optical to microwave quantum interface,” Phys. Rev. Lett. 109, 130503 (2012).
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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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S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89 (2011).
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R. W. Peterson, T. P. Purdy, N. S. Kampel, R.W. Andrews, P. L. Yu, K. W. Lehnert, and C.A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
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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 (2014).
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M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65, 29 (2012).
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O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects, optomechanical superpositions via nested interferometry,” Phys. Rev. Lett. 107, 020405 (2011).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89 (2011).
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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).
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J. B. Clark, F. Lecocq, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Sideband cooling beyond the quantum backaction limit with squeezed light,” Nature 541, 191 (2017).
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M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, and G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107, 023601 (2011).
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V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing an optical pulse as a mechanical excitation in a silica pptomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
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Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “A reversible optical to microwave quantum interface,” Phys. Rev. Lett. 109, 130503 (2012).
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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 (London) 464, 697 (2010).
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M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” Physics 15, 205 (2011).

Blaser, F.

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects, optomechanical superpositions via nested interferometry,” Phys. Rev. Lett. 107, 020405 (2011).
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T. Ojanen and K. Børkje, “Ground state cooling of mechanical motion in the unresolved sideband regime by use of optomechanically induced transparency,” Phys. Rev. A 90, 013824 (2014).
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A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon regime of optomechanics,” Phys. Rev. A 85, 051803 (2012).
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A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
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T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z. Y. Ma, and D. M. Stamper-Kurn, “Tunable cavity optomechanics with ultracold atoms,” Phys. Rev. Lett. 105, 133602 (2010).
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T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z. Y. Ma, and D. M. Stamper-Kurn, “Tunable cavity optomechanics with ultracold atoms,” Phys. Rev. Lett. 105, 133602 (2010).
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M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
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Camarota, B.

M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
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K. Hammerer, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, and T. W. Hansch, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803 (2010).
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Chan, J.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89 (2011).
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M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53 (2018).
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F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
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Chen, P. X.

Chen, Y.

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum Limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
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Chibani, H.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, and G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107, 023601 (2011).
[Crossref] [PubMed]

Cirac, J. I.

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects, optomechanical superpositions via nested interferometry,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

Clark, J. B.

J. B. Clark, F. Lecocq, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Sideband cooling beyond the quantum backaction limit with squeezed light,” Nature 541, 191 (2017).
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Cleland, A. N.

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 (London) 464, 697 (2010).
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Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
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A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 104, 213603 (2010).
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A. A. Clerk, S. M. Girvin, and F. Marquardt, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
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F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
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A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
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Corbitt, T.

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum Limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
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Danilishin, S.

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum Limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
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de Matos Filho, R. L.

R. L. de Matos Filho and W. Vogel, “Second-sideband laser cooling and nonclassical motion of trapped ions,” Phys. Rev. A 50, R1988 (1994).
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Dodonov, V. V.

V. V. Dodonov and S. S. Mizrahi, “Exact stationary photon distributions due to competition between one- and two-photon absorption and emission,” J. Phys. A 30, 5657 (1997).
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F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
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Feng, J. S.

J. S. Feng, L. Tan, H. Q. Gu, and W. M. Liu, “Auxiliary-cavity-assisted ground-state cooling of an optically levitated nanosphere in the unresolved-sideband regime,” Phys. Rev. A 96, 063818 (2017).
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Fiore, V.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing an optical pulse as a mechanical excitation in a silica pptomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref]

Galassi, M.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” Physics 15, 205 (2011).

Genes, C.

K. Hammerer, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, and T. W. Hansch, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803 (2010).
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C. Genes, H. Ritsch, and D. Vitali, “Micromechanical oscillator ground-state cooling via resonant intracavity optical gain or absorption,” Phys. Rev. A 80, 061803 (2009).
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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).
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Ghobadi, R.

B. Pepper, R. Ghobadi, E. Jeffrey, C. Simon, and D. Bouwmeester, “Optomechanical superpositions via nested interferometry,” Phys. Rev. Lett. 109, 023601 (2012).
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Gigan, S.

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).
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Girvin, S. M.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon regime of optomechanics,” Phys. Rev. A 85, 051803 (2012).
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A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

A. A. Clerk, S. M. Girvin, and F. Marquardt, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
[Crossref]

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
[Crossref] [PubMed]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin, and J. G. E. HarrisStrong, “Strong dispersive coupling of a high finesse cavity to a micromechanical membrane,” Nature (London) 452, 72 (2008).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

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

Giuseppe, G. D.

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]

Giuseppe, G. Di.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” Physics 15, 205 (2011).

Gong, Q.

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

Gu, H. Q.

J. S. Feng, L. Tan, H. Q. Gu, and W. M. Liu, “Auxiliary-cavity-assisted ground-state cooling of an optically levitated nanosphere in the unresolved-sideband regime,” Phys. Rev. A 96, 063818 (2017).
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W. J. Gu, Z. Yi, L. H. Sun, and D. H. Xu, “Mechanical cooling in single-photon optomechanics with quadratic nonlinearity,” Phys. Rev. A 92, 023811 (2015).
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Y. Yan, W. J. Gu, and G. X. Li, “Entanglement transfer from two-mode squeezed vacuum light to spatially separated mechanical oscillators via dissipative optomechanical coupling,” Sci. China Phys. Mech. Astron. 58, 050306 (2015).
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Y. Guo, K. Li, W. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (2014).
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Gupta, S.

K. W. Murch, K. L. Moore, and S. Gupta, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561 (2008).
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A. Xuereb, R. Schnabel, and K. Hammerer, “Dissipative optomechanics in a Michelson-Sagnac interferometer,” Phys. Rev. Lett. 107, 213604 (2011).
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Hansch, T. W.

K. Hammerer, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, and T. W. Hansch, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803 (2010).
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Harris, J. G. E.

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 104, 213603 (2010).
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J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
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A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin, and J. G. E. HarrisStrong, “Strong dispersive coupling of a high finesse cavity to a micromechanical membrane,” Nature (London) 452, 72 (2008).
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L. He, Y. X. Liu, S. Yi, C. P. Sun, and F. Nori, “Control of photon propagation via electromagnetically induced transparency in lossless media,” Phys. Rev. A 75, 063818 (2007).
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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 (London) 464, 697 (2010).
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S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
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K. Hammerer, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, and T. W. Hansch, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803 (2010).
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Jayich, A. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
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J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin, and J. G. E. HarrisStrong, “Strong dispersive coupling of a high finesse cavity to a micromechanical membrane,” Nature (London) 452, 72 (2008).
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B. Pepper, R. Ghobadi, E. Jeffrey, C. Simon, and D. Bouwmeester, “Optomechanical superpositions via nested interferometry,” Phys. Rev. Lett. 109, 023601 (2012).
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P. Sekatski, M. Aspelmeyer, and N. Sangouard, “Macroscopic optomechanics from displaced single-photon entanglement,” Phys. Rev. Lett. 112, 080502 (2014).
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A. Xuereb, R. Schnabel, and K. Hammerer, “Dissipative optomechanics in a Michelson-Sagnac interferometer,” Phys. Rev. Lett. 107, 213604 (2011).
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F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
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M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
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R. W. Peterson, T. P. Purdy, N. S. Kampel, R.W. Andrews, P. L. Yu, K. W. Lehnert, and C.A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
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S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80, 688 (1998).
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F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
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H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum Limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
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A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 104, 213603 (2010).
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T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z. Y. Ma, and D. M. Stamper-Kurn, “Tunable cavity optomechanics with ultracold atoms,” Phys. Rev. Lett. 105, 133602 (2010).
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Phys. Rev. X (1)

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

Phys. Today (2)

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65, 29 (2012).
[Crossref]

K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58 (7), 36 (2005).
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Physics (1)

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” Physics 15, 205 (2011).

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A. A. Clerk, S. M. Girvin, and F. Marquardt, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
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Sci. China Phys. Mech. Astron. (2)

Y. Yan, W. J. Gu, and G. X. Li, “Entanglement transfer from two-mode squeezed vacuum light to spatially separated mechanical oscillators via dissipative optomechanical coupling,” Sci. China Phys. Mech. Astron. 58, 050306 (2015).
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Y. C. Liu, Y. F. Xiao, X. S. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 050305 (2015).
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M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
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T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172 (2008).
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Other (1)

L. Qiu, I. Shomroni, P. Seidler, and T. J. Kippenberg, “High-fidelity laser cooling to the quantum ground state of a silicon nanomechanical oscillator,” arXiv:1903.10242 [quant-ph].

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Figures (8)

Fig. 1
Fig. 1 Schematic of the optomechanical system. A membrane oscillator is placed in the middle of cavity 1 which is driven by a continuous-wave input laser, the coupling between them is proportional to the square of position of the oscillator. The cavity2 as an auxiliary cavity only directly couples to the optomechanical cavity 1 with coupling strength J.
Fig. 2
Fig. 2 Energy level diagram of the system in the displaced frame. | n 1 , n 2 , m donates the state of n1 photons in mode a1, n2 photons in mode a2, and m phonons in mode b. The orange double arrow denotes the couplingbetween two cavities with coupling strength J.
Fig. 3
Fig. 3 (a) The optical force spectrum SFF (ω) (in arbitrary units) for different optical coupling strengths J, with Δ1 = 2ωm, κ2 = 0.01ωm. (b) The optical force spectrum SFF (ω) (in arbitrary units) for different detunings of the second cavity κ2 in the optimum conditions with Δ1 = −4ωm, J = 2 6 ω m. The other parameters are taken as Δ2 = −2ωm, κ1 = 10ωm, G = 0.08ωm.
Fig. 4
Fig. 4 The net cooling rate Γopt = AA+ versus the dacay rate of the second cavity κ2 and the coupling strength J. Here the detuning and the decay rate of the first cavity are taken as Δ1 = −4ωm, κ1 = 10ωm, the detuning of the second cavity is taken as Δ2 = −2ωm, and the optomechanical coupling strength is taken as G = 0.08ωm.
Fig. 5
Fig. 5 A comparison of the net cooling rates Γ opt = A A + between the current coupled-cavity optomechanical system and the single-cavity optomechanical system. The former is plotted as the green circle curve under the optimum coupling conditions in Eq. (12) versus the detuning of the auxiliary cavity Δ2 for Δ 1 = 4 ω m, κ 1 = 30 ω m, κ 2 = 0.01 ω m, and the latter is plotted as the blue triangle curve versus Δ1 for κ = 30 ω m. The effective optomechanical coupled strength G = 0.08 ω m is taken.
Fig. 6
Fig. 6 (a) The average phonon number b b of the steady state versus the detuning of the second cavity Δ2 and the optomechanical coupling strength G. The detuning of first cavity Δ1 and the coupling strength between two cavities J meet the optimum condition in Eq. (12). (b) The average phonon number b b versus the effective detuning of the first cavity Δ1 and the coupling strength J. Here the effective detuning of the second cavity Δ 2 = 2 ω m is taken. The white dotted curve satisfies the Eq. (12). The other parameters are taken as κ 1 = 10 ω m, κ 2 = 0.01 ω m, γ m = 10 6 ω m, G = 0.08 ω m and n th = 10.
Fig. 7
Fig. 7 A comparison of the cooling dynamics between the current coupled-cavity optomechanical system and the single-cavity optomechanical system. The numerical results of the average phonon number b b for the coupled-cavity system are plotted with the magenta diamond and green square curves, corresponding to the initial even and odd phonon numbers n th = 10 and 11, respectively, with Δ 1 = 4 ω m, Δ 2 = 2 ω m, J = 2 6 ω m. The numerical results for single-cavity optomechanical system are plotted with orange circle and blue triangle curves corresponding to initial even and odd phonon numbers n th = 10 and 11, respectively, with Δ 1 = 2 ω m. The shadow area denotes the optimal cooling region for b b < 1.
Fig. 8
Fig. 8 Exact numerical result of the average phonon number b b in quadratic optomechanical system for different initial thermal phonon number n th = 10 (green pentagrams), n th = 20 (magenta circles), n th = 30 (blue triangles). The other parameters are taken as Δ 1 = 4 ω m, Δ 2 = 2 ω m, κ 1 = 10 ω m, κ 2 = 0.01 ω m, γ m = 10 6 ω m, J = 2 6 ω m and G = 0.08 ω m. The shadow area denotes the optimal cooling region for b b < 1.

Equations (15)

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H = ω 1 a 1 a 1 + ω 2 a 2 a 2 + ω m b b + g a 1 a 1 ( b + b ) 2 + Ω ( a 1 e i ω L t + a 1 e i ω L t ) + J ( a 1 a 2 + a 1 a 2 ) ,
H = Δ 1 a 1 a 1 + Δ 2 a 2 a 2 + ω m b b + g a 1 a 1 ( b + b ) 2 + Ω ( a 1 + a 1 ) + J ( a 1 a 2 + a 1 a 2 ) ,
α 1 = i Ω ( i Δ 1 κ 1 2 ) + J 2 i Δ 2 κ 2 2 , α 2 = i J α 1 i Δ 2 κ 2 2 , β = 0.
H eff = Δ 1 a 1 a 1 + Δ 2 a 2 a 2 + ω m b b + ( G a 1 + G * a 1 ) ( b + b ) 2 + J ( a 1 a 2 + a 1 a 2 ) ,
H op = Δ 1 a 1 a 1 + Δ 2 a 2 a 2 + J ( a 1 a 2 + a 1 a 2 ) ,
i ω a 1 ( ω ) = ( i Δ 1 κ 1 2 ) a 1 ( ω ) i J a 2 ( ω ) κ 1 a in , 1 ( ω ) , i ω a 2 ( ω ) = ( i Δ 2 κ 2 2 ) a 2 ( ω ) i J a 1 ( ω ) κ 2 a in , 2 ( ω ) ,
a 1 ( ω ) = κ 1 a in , 1 ( ω ) i J χ 2 ( ω ) κ 2 a in , 2 ( ω ) χ ( ω ) ,
χ 1 ( ω ) = 1 κ 1 / 2 i ( ω Δ 1 ) , χ 2 ( ω ) = 1 κ 2 / 2 i ( ω Δ 2 ) , χ ( ω ) = 1 χ 1 ( ω ) + J 2 χ 2 ( ω ) .
S FF ( ω ) = | G | 2 x ZPF 4 [ 1 χ ( ω ) + 1 χ * ( ω ) ] .
P n ˙ = γ m [ n th ( n + 1 ) + ( n th + 1 ) n ] P n + γ m n th n P n 1 + γ m ( n th + 1 ) ( n + 1 ) P n + 1 [ A n ( n 1 ) + A + ( n + 2 ) ( n + 1 ) ] P n + A ( n + 2 ) ( n + 1 ) P n + 2 + A + n ( n 1 ) P n 2 ,
E + = 1 2 ( Δ 1 + Δ 2 + ( Δ 1 + Δ 2 ) 2 + 4 J 2 ) , E = 1 2 ( Δ 1 + Δ 2 ( Δ 1 + Δ 2 ) 2 + 4 J 2 ) .
J = 4 ω m ( 2 ω m Δ 1 ) .
P 2 n + j ( 2 ) = ( 1 m ) m n ( γ + j 1 ) ( 1 ) j 1 ,       j = 0 , 1 ,
P 0 = 1 + 2 ξ 1 + 3 ξ + O ( γ m n t h A ) , P 1 = ξ 1 + 3 ξ + O ( γ m n t h A ) ,   ξ = n t h n t h + 1 ,
ρ ˙ = i [ H eff , ρ ] + κ 1 2 ( 2 a 1 ρ a 1 a 1 a 1 ρ ρ a 1 a 1 ) + κ 2 2 ( 2 a 2 ρ a 2 a 2 a 2 ρ ρ a 2 a 2 ) + γ m 2 ( n th + 1 ) ( 2 b ρ b b b ρ ρ b b ) + γ m 2 n th ( 2 b ρ b b b ρ ρ b b ) ,

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