Abstract

Ground state cooling of mechanical resonator is a way to generate macroscopic quantum states. Here we present a study of optomechanical cooling under the drive of square pulses without smooth profile. By illustrating the dynamical processes of cooling, we show how to choose the amplitudes and durations of square pulses, as well as the intervals between them, so that a mechanical resonator can be quickly cooled down to its ground state. Compared with the cooling under a continuous-wave drive field, the ground state cooling of a mechanical resonator can be performed more efficiently and flexibly by using square pulse drives. At certain times of such cooling process, the thermal phonon number under square pulse drives can become even lower than the theoretical limit for the cooling with a continuous-wave drive field of the same amplitude.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (4)

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 (London) 541, 191–195 (2017).
[Crossref]

B. He, L. Yang, Q. Lin, and M. Xiao, “Radiation pressure cooling as a quantum dynamical process,” Phys. Rev. Lett. 118, 233604 (2017).
[Crossref] [PubMed]

Z.-X. Chen, Q. Lin, B. He, and Z.-Y. Lin, “Entanglement dynamics in double-cavity optomechanical systems,” Opt. Express 25, 17237 (2017).
[Crossref] [PubMed]

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

2016 (3)

Q. Zheng, Y. Yao, and Y. Li, “Optimal quantum parameter estimation in a pulsed quantum optomechanical system,” Phys. Rev. A 93, 013848 (2016).
[Crossref]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802(R) (2016).
[Crossref]

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 (3)

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, Matthew D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Q. Lin and B. He, “Optomechanical entanglement under pulse drive,” Opt. Express 23, 24497 (2015).
[Crossref] [PubMed]

B. He, S.-B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

2014 (3)

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90, 022309 (2014).
[Crossref]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

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

2013 (2)

Y. C. Liu, Y.-F. Xiao, X. Luan, and C.W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref] [PubMed]

Y. C. Liu, Y. W. Hu, C. W. Wong, and Y. F. Xiao, “Review of cavity optomechanical cooling,” Chin. Phys. B,  22, 114213 (2013).
[Crossref]

2012 (5)

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108, 153601 (2012).
[Crossref] [PubMed]

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 2295 (2012).
[Crossref]

E. Verhagen, S. Delèglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85, 063820 (2012).
[Crossref]

2011 (10)

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

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. U.S.A. 108, 16182–16187 (2011).
[Crossref] [PubMed]

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
[Crossref]

R. Riviére, S. Delèglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[Crossref]

J. D. Teufel, T. Donner, D. Li, J. H. 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 (London) 475, 359–363 (2011).
[Crossref]

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 (London) 478, 89–92 (2011).
[Crossref]

M. H. Schleier-Smith, I. D. Leroux, H. Zhang, M. A. VanCamp, and V. Vuletic, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 107, 143005 (2011).
[Crossref] [PubMed]

Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
[Crossref]

Y. Li, L. A. Wu, Y. D. Wang, and L. P. Yang, “Nondeterministic ultrafast ground-state cooling of a mechanical resonator,” Phys. Rev. B 84, 094502 (2011).
[Crossref]

X. T. Wang, S. Vinjanampathy, F. W. Strauch, and K. Jacobs, “Ultraefficient cooling of resonators: beating sideband cooling with quantum control,” Phys. Rev. Lett. 107, 177204 (2011).
[Crossref] [PubMed]

2010 (3)

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, 697–703 (2010).
[Crossref]

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature (London) 463, 72–75 (2010).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010)
[Crossref]

2009 (4)

S. Gröblacher, J. B. Hertzberg, M. R. Vanner, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[Crossref]

A. Schliesser, O. Arcizet, R. Riviére, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[Crossref]

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5, 489–493 (2009).
[Crossref]

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[Crossref]

2008 (7)

J. D. Teufel, J. W. Harlow, C. A. Regal, and K.W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
[Crossref] [PubMed]

G. J. Milburn and M. J. Woolley, “Quantum nanoscience,” Contemp. Phys.,  49, 413–433 (2008).
[Crossref]

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (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]

A. Dantan, C. Genes, D. Vitali, and M. Pinard, “Self-cooling of a movable mirror to the ground state using radiation pressure,” Phys. Rev. A 77, 011804(R) (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] [PubMed]

I. Wilson-Rae, N. Nooshi, J. Dobrindt, T. J. Kippenberg, and W. Zwerger, “Cavity-assisted backaction cooling of mechanical resonators,” New J. Phys. 10, 095007 (2008).
[Crossref]

2007 (2)

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] [PubMed]

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]

2003 (1)

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715–775 (2003).
[Crossref]

2002 (1)

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys. Cond. Mat.,  14, R415–R451 (2002).
[Crossref]

1985 (1)

A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?” Phys. Rev. Lett. 54, 857–860 (1985).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010)
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[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 (London) 478, 89–92 (2011).
[Crossref]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. H. 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 (London) 475, 359–363 (2011).
[Crossref]

Andrews, R.W.

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]

Anetsberger, G.

A. Schliesser, O. Arcizet, R. Riviére, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[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 (London) 464, 697–703 (2010).
[Crossref]

Arcizet, O.

R. Riviére, S. Delèglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[Crossref]

A. Schliesser, O. Arcizet, R. Riviére, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[Crossref]

Aspelmeyer, M.

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

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 2295 (2012).
[Crossref]

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108, 153601 (2012).
[Crossref] [PubMed]

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
[Crossref]

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. U.S.A. 108, 16182–16187 (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 (London) 478, 89–92 (2011).
[Crossref]

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[Crossref]

S. Gröblacher, J. B. Hertzberg, M. R. Vanner, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (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]

Aumentado, J.

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 (London) 541, 191–195 (2017).
[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 (London) 464, 697–703 (2010).
[Crossref]

Brukner, C.

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. U.S.A. 108, 16182–16187 (2011).
[Crossref] [PubMed]

Cerrillo, J.

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108, 153601 (2012).
[Crossref] [PubMed]

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[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 (London) 478, 89–92 (2011).
[Crossref]

Chen, J. P.

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]

Chen, Z.-X.

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. H. 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 (London) 475, 359–363 (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 (London) 541, 191–195 (2017).
[Crossref]

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, 697–703 (2010).
[Crossref]

Clerk, A. A.

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature (London) 463, 72–75 (2010).
[Crossref]

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (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]

Cohen, J. D.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, Matthew D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Cole, G. D.

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 2295 (2012).
[Crossref]

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. U.S.A. 108, 16182–16187 (2011).
[Crossref] [PubMed]

Dantan, A.

A. Dantan, C. Genes, D. Vitali, and M. Pinard, “Self-cooling of a movable mirror to the ground state using radiation pressure,” Phys. Rev. A 77, 011804(R) (2008).
[Crossref]

Delèglise, S.

E. Verhagen, S. Delèglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

R. Riviére, S. Delèglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[Crossref]

Dobrindt, J.

I. Wilson-Rae, N. Nooshi, J. Dobrindt, T. J. Kippenberg, and W. Zwerger, “Cavity-assisted backaction cooling of mechanical resonators,” New J. Phys. 10, 095007 (2008).
[Crossref]

Dobrindt, J. M.

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

Donner, T.

J. D. Teufel, T. Donner, D. Li, J. H. 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 (London) 475, 359–363 (2011).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise, Chap. 5.1, Springer Verlag, Berlin, Heidelberg (2000).
[Crossref]

Garg, A.

A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?” Phys. Rev. Lett. 54, 857–860 (1985).
[Crossref] [PubMed]

Gavartin, E.

R. Riviére, S. Delèglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[Crossref]

Genes, C.

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[Crossref]

A. Dantan, C. Genes, D. Vitali, and M. Pinard, “Self-cooling of a movable mirror to the ground state using radiation pressure,” Phys. Rev. A 77, 011804(R) (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]

Ghobadi, R.

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90, 022309 (2014).
[Crossref]

Gigan, S.

S. Gröblacher, J. B. Hertzberg, M. R. Vanner, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (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]

Girvin, S. M.

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (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]

Gröblacher, S.

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 (London) 478, 89–92 (2011).
[Crossref]

S. Gröblacher, J. B. Hertzberg, M. R. Vanner, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[Crossref]

Hammerer, K.

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. U.S.A. 108, 16182–16187 (2011).
[Crossref] [PubMed]

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
[Crossref]

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[Crossref]

Harlow, J. H.

J. D. Teufel, T. Donner, D. Li, J. H. 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 (London) 475, 359–363 (2011).
[Crossref]

Harlow, J. W.

J. D. Teufel, J. W. Harlow, C. A. Regal, and K.W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
[Crossref] [PubMed]

He, B.

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

B. He, L. Yang, Q. Lin, and M. Xiao, “Radiation pressure cooling as a quantum dynamical process,” Phys. Rev. Lett. 118, 233604 (2017).
[Crossref] [PubMed]

Z.-X. Chen, Q. Lin, B. He, and Z.-Y. Lin, “Entanglement dynamics in double-cavity optomechanical systems,” Opt. Express 25, 17237 (2017).
[Crossref] [PubMed]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802(R) (2016).
[Crossref]

B. He, S.-B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

Q. Lin and B. He, “Optomechanical entanglement under pulse drive,” Opt. Express 23, 24497 (2015).
[Crossref] [PubMed]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90, 022309 (2014).
[Crossref]

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85, 063820 (2012).
[Crossref]

Hertzberg, J. B.

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature (London) 463, 72–75 (2010).
[Crossref]

S. Gröblacher, J. B. Hertzberg, M. R. Vanner, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[Crossref]

Hill, J. T.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[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 (London) 478, 89–92 (2011).
[Crossref]

Hofer, J.

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 2295 (2012).
[Crossref]

Hofer, S. G.

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
[Crossref]

Hofheinz, 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 (London) 464, 697–703 (2010).
[Crossref]

Hu, Y. W.

Y. C. Liu, Y. W. Hu, C. W. Wong, and Y. F. Xiao, “Review of cavity optomechanical cooling,” Chin. Phys. B,  22, 114213 (2013).
[Crossref]

Huang, S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010)
[Crossref]

Jacobs, K.

X. T. Wang, S. Vinjanampathy, F. W. Strauch, and K. Jacobs, “Ultraefficient cooling of resonators: beating sideband cooling with quantum control,” Phys. Rev. Lett. 107, 177204 (2011).
[Crossref] [PubMed]

Kampel, N. S.

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]

Kim, M. S.

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. U.S.A. 108, 16182–16187 (2011).
[Crossref] [PubMed]

Kippenberg, T. J.

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

E. Verhagen, S. Delèglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

R. Riviére, S. Delèglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[Crossref]

A. Schliesser, O. Arcizet, R. Riviére, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[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] [PubMed]

I. Wilson-Rae, N. Nooshi, J. Dobrindt, T. J. Kippenberg, and W. Zwerger, “Cavity-assisted backaction cooling of mechanical resonators,” New J. Phys. 10, 095007 (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] [PubMed]

Krause, A.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[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 (London) 478, 89–92 (2011).
[Crossref]

Law, C. K.

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

Lecocq, F.

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 (London) 541, 191–195 (2017).
[Crossref]

Leggett, A. J.

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys. Cond. Mat.,  14, R415–R451 (2002).
[Crossref]

A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?” Phys. Rev. Lett. 54, 857–860 (1985).
[Crossref] [PubMed]

Lehnert, K.W.

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]

J. D. Teufel, T. Donner, D. Li, J. H. 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 (London) 475, 359–363 (2011).
[Crossref]

J. D. Teufel, J. W. Harlow, C. A. Regal, and K.W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
[Crossref] [PubMed]

Lenander, 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 (London) 464, 697–703 (2010).
[Crossref]

Leroux, I. D.

M. H. Schleier-Smith, I. D. Leroux, H. Zhang, M. A. VanCamp, and V. Vuletic, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 107, 143005 (2011).
[Crossref] [PubMed]

Li, D.

J. D. Teufel, T. Donner, D. Li, J. H. 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 (London) 475, 359–363 (2011).
[Crossref]

Li, Y.

Q. Zheng, Y. Yao, and Y. Li, “Optimal quantum parameter estimation in a pulsed quantum optomechanical system,” Phys. Rev. A 93, 013848 (2016).
[Crossref]

Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
[Crossref]

Y. Li, L. A. Wu, Y. D. Wang, and L. P. Yang, “Nondeterministic ultrafast ground-state cooling of a mechanical resonator,” Phys. Rev. B 84, 094502 (2011).
[Crossref]

Liao, J.-Q.

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

Lin, Q.

B. He, L. Yang, Q. Lin, and M. Xiao, “Radiation pressure cooling as a quantum dynamical process,” Phys. Rev. Lett. 118, 233604 (2017).
[Crossref] [PubMed]

Q. Lin, B. He, and M. Xiao, “Mass sensing by detecting the quadrature of a coupled light field,” Phys. Rev. A 96, 043812 (2017).
[Crossref]

Z.-X. Chen, Q. Lin, B. He, and Z.-Y. Lin, “Entanglement dynamics in double-cavity optomechanical systems,” Opt. Express 25, 17237 (2017).
[Crossref] [PubMed]

Q. Lin and B. He, “Optomechanical entanglement under pulse drive,” Opt. Express 23, 24497 (2015).
[Crossref] [PubMed]

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90, 022309 (2014).
[Crossref]

Lin, Z.-Y.

Liu, Y. C.

Y. C. Liu, Y.-F. Xiao, X. Luan, and C.W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref] [PubMed]

Y. C. Liu, Y. W. Hu, C. W. Wong, and Y. F. Xiao, “Review of cavity optomechanical cooling,” Chin. Phys. B,  22, 114213 (2013).
[Crossref]

Luan, X.

Y. C. Liu, Y.-F. Xiao, X. Luan, and C.W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref] [PubMed]

Lucero, E.

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, 697–703 (2010).
[Crossref]

MacCabe, G. S.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, Matthew D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Machnes, S.

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108, 153601 (2012).
[Crossref] [PubMed]

Macklin, C.

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature (London) 463, 72–75 (2010).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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X. T. Wang, S. Vinjanampathy, F. W. Strauch, and K. Jacobs, “Ultraefficient cooling of resonators: beating sideband cooling with quantum control,” Phys. Rev. Lett. 107, 177204 (2011).
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M. H. Schleier-Smith, I. D. Leroux, H. Zhang, M. A. VanCamp, and V. Vuletic, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 107, 143005 (2011).
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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, 697–703 (2010).
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E. Verhagen, S. Delèglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
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S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108, 153601 (2012).
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B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
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Y. C. Liu, Y.-F. Xiao, X. Luan, and C.W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
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M. H. Schleier-Smith, I. D. Leroux, H. Zhang, M. A. VanCamp, and V. Vuletic, “Optomechanical cavity cooling of an atomic ensemble,” Phys. Rev. Lett. 107, 143005 (2011).
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A. Schliesser, O. Arcizet, R. Riviére, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
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Figures (6)

Fig. 1
Fig. 1 Model of pulsed optomechanical cooling. The pulse drive field with red detuning Δ = ωm is sent into the cavity with the one movable mirror (mechanical resonator), which will be cooled down to its ground state given a proper profile of the pulses.
Fig. 2
Fig. 2 Evolution of phonon number under a single square pulse drive field with different drive intensity E/κ and duration κt0, together with that under a single Gaussian pulse drive field. For comparison, the product (E/κ) × (κt0) is set to be fixed as 107. The mechanical resonator is seen to be quickly cooled down with strong intensity, but the final phonon number will oscillate due to a stronger pulse amplitude. The other parameters are gm/κ = 10−4, Δ/κ = ωm/κ = 100, γm/κ = 10−3, and nth = 100.
Fig. 3
Fig. 3 Evolution of phonon number under multiple pulses. The duration of and interval between the pulses are set to be equal. The ground state cooling can be achieved and the ground state can be preserved with the duration κt0 = 2 and the intensities E/κ = 8×105, 106, while the ground state cannot be well preserved given the longer durations κt0 = 10, 5 or cannot be achieved given the relatively weak intensity E/κ = 5 × 105. The other parameters are chosen as gm/κ = 10−4, Δ/κ = ωm/κ = 100, and γm/κ = 10−3.
Fig. 4
Fig. 4 Relations between the time of reaching the first dip (blue circle) and the corresponding phonon number (red star) with the dimensionless effective intensity J = (gm/ωm) × (E/κ). Increasing the effective intensity will increase the cooling speed (a shorter time of first dip), but the corresponding phonon number will become oscillating with the parameter. The system parameters are set as gm/κ = 10−4, Δ/κ = ωm/κ = 100, γm/κ = 10−3, and nth = 100.
Fig. 5
Fig. 5 Evolution of phonon number with the short duration κt1 = 0.34 and different time interval κt2 = 0.34, 1.0, 1.5, 2.0, 2.5, 3.0. The intensity is strong as E/κ = 5 × 106. The dash-dotted line denotes the theoretical cooling limit nmκ/(γmnth) = 1 with continuous-wave drive field, which can be beyond by the pulsed optomechanical cooling. The ellipses mark the periods, in which the mechanical resonator is heated up when the pulses are applied. All other parameters are gm/κ = 10−4, Δ/κ = ωm/κ = 100, γm/κ = 10−3, and nth = 100.
Fig. 6
Fig. 6 Comparisons of the mechanical resonator’s displacements Xm(t) predicted with the nonlinear dynamical equations Eq. (12) (blue) and our linearized equations Eq. (7) (red). Theses comparisons are made with some different parameters ωm/κ and J = (gm/ωm) × (E/κ). The common parameters for the different situations are chosen as gm/κ = 10−4, Δ/κ = ωm/κ, and γm/κ = 10−3.

Equations (12)

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H ( t ) = ω c a ^ a ^ + ω m b ^ b ^ + i [ a ^ E ( t ) e i ω L t a ^ E * ( t ) e i ω L t ] H S ( t ) g m a ^ a ^ ( b ^ + b ^ ) H OM + i 2 κ { a ^ ξ ^ c ( t ) a ^ ξ ^ c ( t ) } + i 2 γ m { b ^ ξ ^ m ( t ) b ^ ξ ^ m ( t ) } H SR .
U ( t ) = 𝒯 exp { i 0 t d τ H S ( τ ) } × 𝒯 exp { i 0 t d τ [ H eff ( τ ) + H N ( τ ) ] } ,
𝒯 exp { i 0 t d τ H ^ S ( τ ) } a ^ 𝒯 exp { i 0 t d τ H ^ S ( τ ) } = e i ω c t ( a ^ + 0 t d τ E ( τ ) e i Δ τ ) e i ω c t ( a ^ + F ( t ) ) ,
𝒯 exp { i 0 t d τ H ^ S ( τ ) } b ^ 𝒯 exp { i 0 t d τ H ^ S ( τ ) } = e i ω m t b ^ ,
H eff ( τ ) = g m [ F ( τ ) a ^ + F * ( τ ) a ^ + | E ( τ ) | 2 ] ( e i ω m τ b ^ + e i ω m τ b ^ ) + i 2 κ { e i ω c τ ( a ^ + F * ( τ ) ) ξ ^ c ( τ ) e i ω c τ ( a ^ + F ( τ ) ) ξ ^ c ( τ ) } + i 2 γ m ( e i ω m τ b ^ ξ ^ m ( τ ) e i ω m τ b ^ ξ ^ m ( τ ) ) ,
H N ( τ ) = g m a ^ a ^ ( e i ω m τ b ^ + e i ω m τ b ^ ) .
a ^ ˙ = κ a ^ + i g m F ( t ) ( e i ω m t b ^ + e i ω m t b ^ ) κ F ( t ) + 2 κ e i ω c t ξ ^ c ( t ) , b ^ ˙ = γ m b ^ + i g m e i ω m t [ F ( t ) a ^ + F * ( t ) a ^ ] + i g m e i ω m t | F ( t ) | 2 + 2 γ m e i ω m t ξ ^ m ( t ) ,
n m ( t ) = ( b ^ ( t ) b ^ ( t ) ) δ b ^ ( t ) ( b ^ ( t ) b ^ ( t ) ) δ b ^ ( t ) = b ^ b ^ ( t ) b ^ ( t ) b ^ ( t )
F ( t ) = { i E / Δ ( 1 e i Δ t ) ; t t 0 , 0 ; t > t 0 .
F ( t ) = { i E Δ ( 1 e i Δ ( t j ( t 1 + t 2 ) ) ) ; 0 < t j ( t 1 + t 2 ) t 1 , 0 ; 0 < t j ( t 1 + t 2 ) t 1 t 2 ,
Δ n m = 2 γ m n th Δ t ,
a ^ ˙ = κ a ^ + i g m ( e i ω m t b ^ + e i ω m t b ^ ) a ^ + E e i Δ t + 2 κ ξ ^ c ( t ) b ^ ˙ = γ m b ^ + i g m e i ω m t a ^ a ^ + 2 γ m ξ ^ m ( t ) .

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