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Cavity optomechanics becomes a promising field in quantum and nano technologies. Motivated by the optomechancial experiment with the membrane located in a high-finesse optical cavity and theoretical treatment on two membranes cavity optomechanics, we here study the optomechanical interaction of the system consisting of triple membranes within an optical cavity. The increase of membranes will increase the normal modes of the cavity and mechanical fields, and thus enrich the forms of optomechanical interaction. Firstly, we use the transfer matrix and resonance transmission methods to obtain the dispersion relation between the eigen-frequencies of the optical modes and the mechanical motions. Owing to the existence of triple mechanical membranes, the system possesses different forms of collective mechanical motion, and here we focus on the center-of-mass (COM) motion and relative motion of the equally placed membranes. The numerical solutions of the dispersion relation show that the optical eigenmodes are comprised of a group of closely spaced avoided-crossing quaternion of wave numbers, which arise from the transmission and reflection of the optical field at the membranes and the tunneling couplings between subcavity modes. Moreover, the change of each eigen wave number along each form of the mechanical motion is different, which implies the different forms of optomechanical coupling between eigenmodes and mechanical motions. Then, to achieve the explicit expressions of the optomechanical coupling, it is sufficient to use the perturbation method under the equilibrium condition of the system, where the amplitude of mechanical motion is much smaller than the optical wavelength. With using the implicit function differentiation theorem, the optomechanical coupling strengths between the four optical modes and the COM and relative mechanical motions are obtained respectively. We find that the strong quadratic optomechanical coupling between the optical modes and COM motion can be achieved, and the linear and quadratic couplings between the optical modes and relative motion can both be realized. By tuning the laser to pump different optical modes, we can choose either the linear or the quadratic coupling to the relative motion. Our method is universal to multi-membrane system, and the results may provide some references to theoretical and experimental investigations on the multi-membrane cavity optomechanics.
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Keywords:
- multi-membrane cavity optomechanics /
- collective mechanical motion /
- dispersion relation /
- optomechanical coupling strength
[1] Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391
[2] Kippenberg T J, Vahala K J 2007 Opt. Express 15 17172
[3] Chen X, Liu X W, Zhang K Y, Yuan C H, Zhang W P 2015 Acta Phys. Sin. 64 164211 (in Chinese) [陈雪, 刘晓威, 张可烨, 袁春华, 张卫平 2015 物理学报 64 164211]
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[7] Carmon T, Rokhsari H, Yang L, Kippenberg T J, Vahala K J 2005 Phys. Rev. Lett. 94 223902
[8] Rokhsari H, Kippenberg T J, Carmon T, Vahala K J 2005 Opt. Express 13 5293
[9] Thompson J D, Zwickl B M, Jayich A M, Marquardt F, Girvin S M, Harris J G E 2008 Nature 452 72
[10] Anetsberger G, Arcizet O, Unterreithmeier Q P, Riviere R, Schliesser A, Weig E M, Kotthaus J P, Kippenberg T J 2009 Nat. Phys. 5 909
[11] Lee D, Underwood M, Mason D, Shkarin A B, Hoch S W, Harris J G E 2015 Nat. Commun. 6 6232
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[13] Ludwig M, Hammerer K, Marquardt F 2010 Phys. Rev. A 82 012333
[14] Komar P, Bennett S D, Stannigel K, Habraken S J M, Rabl P, Zoller P, Lukin M D 2013 Phys. Rev. A 87 013839
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[16] Tian L 2013 Phys. Rev. Lett. 110 233602
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[18] Bui C H, Zheng J, Hoch S W, Lee L Y T, Harris J G E, Wong C W 2012 Appl. Phys. Lett. 100 021110
[19] Sankey J C, Yang C, Zwickl B M, Jayich A M, Harris J G E 2010 Nature Phys. 6 707
[20] Jacobs N E F, Hoch S W, Sankey J C, Kashkanova A, Jayich A M, Deutsch C, Reichel J, Harris J G E 2012 Appl. Phys. Lett. 101 221109
[21] Underwood M, Mason D, Lee D, Xu H, Jiang L, Shkarin A B, Borkje K, Girvin S M, Harris J G E 2015 Phys. Rev. A 92 061801
[22] Jayich A M, Sankey J C, Zwickl B M, Yang C, Thompson J D, Girvin S M, Clerk A A, Marquardt F, Harris J G E 2008 New J. Phys. 10 095008
[23] Bhattacharya M, Meystre 2008 Phys. Rev. A 78 041801
[24] Xuereb A, Genes C, Dantan A 2012 Phys. Rev. Lett. 109 223601
[25] Tomadin A, Diehl S, Lukin M D, Rabl P, Zoller P 2012 Phys. Rev. A 86 033821
[26] Ludwig M, Marquardt F 2013 Phys. Rev. Lett. 111 073603
[27] Seok H, Buchmann L F, Wright E M, Meystre P 2013 Phys. Rev. A 88 063850
[28] Xuereb A, Genes C, Pupillo G, Paternostro M, Dantan A 2014 Phys. Rev. Lett. 112 133604
[29] Stannigel K, Komar P, Habraken S J M, Bennett S D, Lukin M D, Zoller P, Rabl P 2012 Phys. Rev. Lett. 109 013603
[30] Bhattacharya M, Uys H, Meystre P 2008 Phys. Rev. A 77 033819
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[32] Zhang X, Zhang L 2015 Sci. Sin.: Phys. Mech. Astron. 45 044201 (in Chinese) [张旭, 张林 2015 中国科学: 物理学 力学 天文学 45 044201]
[33] Fader W J 1985 IEEE J. Quantum Electron. 21 1838
[34] Xu X W, Zhao Y J, Liu Y X 2013 Phys. Rev. A 88 022325
[35] Xuereb A, Genes C, Dantan A 2013 Phys. Rev. A 88 053803
[36] Deutsch I H, Spreeuw R J C, Rolston S L, Phillips W D 1995 Phys. Rev. A 52 1394
[37] Ludwig M, Safavi-Naeini A H, Painter O, Marquardt F 2012 Phys. Rev. Lett. 109 063601
[38] Gu W, Yi Z, Sun L, Xu D 2015 Phys. Rev. A 92 023811
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[1] Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391
[2] Kippenberg T J, Vahala K J 2007 Opt. Express 15 17172
[3] Chen X, Liu X W, Zhang K Y, Yuan C H, Zhang W P 2015 Acta Phys. Sin. 64 164211 (in Chinese) [陈雪, 刘晓威, 张可烨, 袁春华, 张卫平 2015 物理学报 64 164211]
[4] Teufel J D, Donner T, Li D, Harlow J W, Allman M S, Cicak K, Sirois A J, Whittaker J D, Lehnert K W, Simmonds R W 2011 Nature 475 359
[5] Connell A D, Hofheinz M, Ansmann M, Bialczak R C, Lenander M, Lucero E, Neeley M, Sank D, Wang H, Weides M, Wenner J, Martinis J M, Cleland A N 2010 Nature 464 697
[6] Liu Y C, Hu Y W, Wong C W, Xiao Y F 2013 Chin. Phys. B 22 114213
[7] Carmon T, Rokhsari H, Yang L, Kippenberg T J, Vahala K J 2005 Phys. Rev. Lett. 94 223902
[8] Rokhsari H, Kippenberg T J, Carmon T, Vahala K J 2005 Opt. Express 13 5293
[9] Thompson J D, Zwickl B M, Jayich A M, Marquardt F, Girvin S M, Harris J G E 2008 Nature 452 72
[10] Anetsberger G, Arcizet O, Unterreithmeier Q P, Riviere R, Schliesser A, Weig E M, Kotthaus J P, Kippenberg T J 2009 Nat. Phys. 5 909
[11] Lee D, Underwood M, Mason D, Shkarin A B, Hoch S W, Harris J G E 2015 Nat. Commun. 6 6232
[12] Shkarin A B, Jacobs N E F, Hoch S W, Kashkanova A D, Deutsch C, Reichel J, Harris J G E 2014 Phys. Rev. Lett. 112 013602
[13] Ludwig M, Hammerer K, Marquardt F 2010 Phys. Rev. A 82 012333
[14] Komar P, Bennett S D, Stannigel K, Habraken S J M, Rabl P, Zoller P, Lukin M D 2013 Phys. Rev. A 87 013839
[15] Verlot P, Tavernarakis A, Briant T, Cohadon P F, Heidmann A 2009 Phys. Rev. Lett. 102 103601
[16] Tian L 2013 Phys. Rev. Lett. 110 233602
[17] Andrews R W, Peterson R W, Purdy T P, Cicak K, Simmonds R W, Regal C A, Lehnert K W 2014 Nat. Phys. 10 321
[18] Bui C H, Zheng J, Hoch S W, Lee L Y T, Harris J G E, Wong C W 2012 Appl. Phys. Lett. 100 021110
[19] Sankey J C, Yang C, Zwickl B M, Jayich A M, Harris J G E 2010 Nature Phys. 6 707
[20] Jacobs N E F, Hoch S W, Sankey J C, Kashkanova A, Jayich A M, Deutsch C, Reichel J, Harris J G E 2012 Appl. Phys. Lett. 101 221109
[21] Underwood M, Mason D, Lee D, Xu H, Jiang L, Shkarin A B, Borkje K, Girvin S M, Harris J G E 2015 Phys. Rev. A 92 061801
[22] Jayich A M, Sankey J C, Zwickl B M, Yang C, Thompson J D, Girvin S M, Clerk A A, Marquardt F, Harris J G E 2008 New J. Phys. 10 095008
[23] Bhattacharya M, Meystre 2008 Phys. Rev. A 78 041801
[24] Xuereb A, Genes C, Dantan A 2012 Phys. Rev. Lett. 109 223601
[25] Tomadin A, Diehl S, Lukin M D, Rabl P, Zoller P 2012 Phys. Rev. A 86 033821
[26] Ludwig M, Marquardt F 2013 Phys. Rev. Lett. 111 073603
[27] Seok H, Buchmann L F, Wright E M, Meystre P 2013 Phys. Rev. A 88 063850
[28] Xuereb A, Genes C, Pupillo G, Paternostro M, Dantan A 2014 Phys. Rev. Lett. 112 133604
[29] Stannigel K, Komar P, Habraken S J M, Bennett S D, Lukin M D, Zoller P, Rabl P 2012 Phys. Rev. Lett. 109 013603
[30] Bhattacharya M, Uys H, Meystre P 2008 Phys. Rev. A 77 033819
[31] Hartmann M J, Plenio M B 2008 Phys. Rev. Lett. 101 200503
[32] Zhang X, Zhang L 2015 Sci. Sin.: Phys. Mech. Astron. 45 044201 (in Chinese) [张旭, 张林 2015 中国科学: 物理学 力学 天文学 45 044201]
[33] Fader W J 1985 IEEE J. Quantum Electron. 21 1838
[34] Xu X W, Zhao Y J, Liu Y X 2013 Phys. Rev. A 88 022325
[35] Xuereb A, Genes C, Dantan A 2013 Phys. Rev. A 88 053803
[36] Deutsch I H, Spreeuw R J C, Rolston S L, Phillips W D 1995 Phys. Rev. A 52 1394
[37] Ludwig M, Safavi-Naeini A H, Painter O, Marquardt F 2012 Phys. Rev. Lett. 109 063601
[38] Gu W, Yi Z, Sun L, Xu D 2015 Phys. Rev. A 92 023811
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