Abstract

In this paper, the non-Markovian dynamics of an optomechanical system is analyzed by using the non-Markovian quantum state diffusion (NMQSD) method. An exact solution is obtained for the system composed of a Fabry-Pérot (F-P) cavity with two movable mirrors without the linearization of the Hamiltonian. Based on the solution, we find that the memory effect of the non-Markovian environment can be used to generate macroscopic entanglement between two mirrors. In order to achieve the maximum entanglement generation, the non-Markovian properties of the environment have to be chosen carefully depending on the properties of the system. Then, we also show that the coherence (superposition) in the initial state may produce entanglement in the evolution. At last, we show the entanglement sudden death and revival significantly depend on the strength of the memory effect, and the entanglement revival can be only observed in non-Markovian case. Our treatment, as an example, paves a way to exactly solve a large category of optomechanical systems without the linearization of the Hamiltonian.

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

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2019 (1)

2017 (1)

2016 (1)

W.-Z. Zhang, J. Cheng, W.-D. Li, and L. Zhou, “Optomechanical cooling in the non-markovian regime,” Phys. Rev. A 93(6), 063853 (2016).
[Crossref]

2015 (1)

S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, “Observation of non-Markovian micromechanical Brownian motion,” Nat. Commun. 6(1), 7606 (2015).
[Crossref]

2014 (7)

X. Zhao, J. Jing, J. You, and T. Yu, “Dynamics of coupled cavity arrays embedded in a non-markovian bath,” Quantum Inf. & Comput. 14, 741–756 (2014).

J. Xu, X. Zhao, J. Jing, L.-A. Wu, and T. Yu, “Perturbation methods for the non-markovian quantum state diffusion equation,” J. Phys. A: Math. Theor. 47(43), 435301 (2014).
[Crossref]

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quantum non-markovianity: characterization, quantification and detection,” Rep. Prog. Phys. 77(9), 094001 (2014).
[Crossref]

P. Sekatski, M. Aspelmeyer, and N. Sangouard, “Macroscopic optomechanics from displaced single-photon entanglement,” Phys. Rev. Lett. 112(8), 080502 (2014).
[Crossref]

K. Zhang, F. Bariani, and P. Meystre, “Theory of an optomechanical quantum heat engine,” Phys. Rev. A 90(2), 023819 (2014).
[Crossref]

R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112(8), 080503 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

2013 (9)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
[Crossref]

Y. Chen, “Macroscopic quantum mechanics: theory and experimental concepts of optomechanics,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104001 (2013).
[Crossref]

L. Zhou, J. Cheng, Y. Han, and W. Zhang, “Nonlinearity enhancement in optomechanical systems,” Phys. Rev. A 88(6), 063854 (2013).
[Crossref]

W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated-emission laser,” Phys. Rev. A 88(2), 022338 (2013).
[Crossref]

W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated emission laser via cascade-driven coherence,” Phys. Rev. A 88(5), 052301 (2013).
[Crossref]

W. Shi, X. Zhao, and T. Yu, “Non-markovian fermionic stochastic schrödinger equation for open system dynamics,” Phys. Rev. A 87(5), 052127 (2013).
[Crossref]

M. Chen and J. Q. You, “Non-markovian quantum state diffusion for an open quantum system in fermionic environments,” Phys. Rev. A 87(5), 052108 (2013).
[Crossref]

J. Jing, X. Zhao, J. Q. You, W. T. Strunz, and T. Yu, “Many-body quantum trajectories of non-markovian open systems,” Phys. Rev. A 88(5), 052122 (2013).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gröblacher, H. Miao, Y. Chen, M. Aspelmeyer, and O. Painter, “Laser noise in cavity-optomechanical cooling and thermometry,” New J. Phys. 15(3), 035007 (2013).
[Crossref]

2012 (7)

W.-M. Zhang, P.-Y. Lo, H.-N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109(17), 170402 (2012).
[Crossref]

S. Olivares, “Quantum optics in the phase space,” Eur. Phys. J.: Spec. Top. 203(1), 3–24 (2012).
[Crossref]

X. Zhao, W. Shi, L.-A. Wu, and T. Yu, “Fermionic stochastic schrödinger equation and master equation: An open-system model,” Phys. Rev. A 86(3), 032116 (2012).
[Crossref]

J. Jing, X. Zhao, J. Q. You, and T. Yu, “Time-local quantum-state-diffusion equation for multilevel quantum systems,” Phys. Rev. A 85(4), 042106 (2012).
[Crossref]

A. Mari and J. Eisert, “Cooling by heating: Very hot thermal light can significantly cool quantum systems,” Phys. Rev. Lett. 108(12), 120602 (2012).
[Crossref]

B. Cleuren, B. Rutten, and C. Van den Broeck, “Cooling by heating: Refrigeration powered by photons,” Phys. Rev. Lett. 108(12), 120603 (2012).
[Crossref]

K. Stannigel, P. Komar, S. Habraken, S. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref]

2011 (7)

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

J. Teufel, D. Li, M. Allman, K. Cicak, A. Sirois, J. Whittaker, and R. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[Crossref]

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. 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(7356), 359–363 (2011).
[Crossref]

J. Chan, T. 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(7367), 89–92 (2011).
[Crossref]

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from markovian to non-markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

X. Zhao, J. Jing, B. Corn, and T. Yu, “Dynamics of interacting qubits coupled to a common bath: Non-markovian quantum-state-diffusion approach,” Phys. Rev. A 84(3), 032101 (2011).
[Crossref]

2010 (3)

J. Jing and T. Yu, “Non-markovian relaxation of a three-level system: Quantum trajectory approach,” Phys. Rev. Lett. 105(24), 240403 (2010).
[Crossref]

A. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-markovianity of quantum evolutions,” Phys. Rev. Lett. 105(5), 050403 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

2009 (5)

F. M. S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

H.-P. Breuer, E.-M. Laine, and J. Piilo, “Measure for the degree of non-markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103(21), 210401 (2009).
[Crossref]

X.-Y. Zhao, Y.-H. Ma, and L. Zhou, “Generation of multi-mode-entangled light,” Opt. Commun. 282(8), 1593–1597 (2009).
[Crossref]

T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323(5914), 598–601 (2009).
[Crossref]

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, and B. M. Garraway, “Pseudomodes as an effective description of memory: Non-markovian dynamics of two-state systems in structured reservoirs,” Phys. Rev. A 80(1), 012104 (2009).
[Crossref]

2008 (2)

Z. Ficek and R. Tanaś, “Delayed sudden birth of entanglement,” Phys. Rev. A 77(5), 054301 (2008).
[Crossref]

M. W. Y. Tu and W.-M. Zhang, “Non-markovian decoherence theory for a double-dot charge qubit,” Phys. Rev. B 78(23), 235311 (2008).
[Crossref]

2007 (3)

J.-H. An and W.-M. Zhang, “Non-markovian entanglement dynamics of noisy continuous-variable quantum channels,” Phys. Rev. A 76(4), 042127 (2007).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. Böhm, 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(3), 030405 (2007).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[Crossref]

2006 (3)

L. Zhou, H. Xiong, and M. S. Zubairy, “Single atom as a macroscopic entanglement source,” Phys. Rev. A 74(2), 022321 (2006).
[Crossref]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

T. Yu and J. H. Eberly, “Quantum open system theory: Bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
[Crossref]

2005 (3)

H. Xiong, M. Scully, and M. Zubairy, “Correlated spontaneous emission laser as an entanglement amplifier,” Phys. Rev. Lett. 94(2), 023601 (2005).
[Crossref]

F.-L. Li, H. Xiong, and M. S. Zubairy, “Coherence-induced entanglement; (R),” Phys. Rev. A 72(1), 010303 (2005).
[Crossref]

K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58(7), 36–42 (2005).
[Crossref]

2004 (4)

W. Dür and H.-J. Briegel, “Stability of macroscopic entanglement under decoherence,” Phys. Rev. Lett. 92(18), 180403 (2004).
[Crossref]

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70(2), 022318 (2004).
[Crossref]

W. T. Strunz and T. Yu, “Convolutionless non-markovian master equations and quantum trajectories: Brownian motion,” Phys. Rev. A 69(5), 052115 (2004).
[Crossref]

T. Yu, “Non-markovian quantum trajectories versus master equations: Finite-temperature heat bath,” Phys. Rev. A 69(6), 062107 (2004).
[Crossref]

2003 (1)

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91(13), 130401 (2003).
[Crossref]

2002 (1)

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88(12), 120401 (2002).
[Crossref]

2001 (1)

F. Plastina, R. Fazio, and G. M. Palma, “Macroscopic entanglement in josephson nanocircuits,” Phys. Rev. B 64(11), 113306 (2001).
[Crossref]

2000 (2)

R. Simon, “Peres-horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2726–2729 (2000).
[Crossref]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

1999 (3)

W. T. Strunz, L. Diósi, and N. Gisin, “Open system dynamics with non-markovian quantum trajectories,” Phys. Rev. Lett. 82(9), 1801–1805 (1999).
[Crossref]

T. Yu, L. Diósi, N. Gisin, and W. T. Strunz, “Non-markovian quantum-state diffusion: Perturbation approach,” Phys. Rev. A 60(1), 91–103 (1999).
[Crossref]

S. Bose, K. Jacobs, and P. L. Knight, “Scheme to probe the decoherence of a macroscopic object,” Phys. Rev. A 59(5), 3204–3210 (1999).
[Crossref]

1998 (2)

S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80(4), 688–691 (1998).
[Crossref]

L. Diósi, N. Gisin, and W. T. Strunz, “Non-markovian quantum state diffusion,” Phys. Rev. A 58(3), 1699–1712 (1998).
[Crossref]

1997 (1)

S. Mancini, V. Man’ko, and P. Tombesi, “Ponderomotive control of quantum macroscopic coherence,” Phys. Rev. A 55(4), 3042–3050 (1997).
[Crossref]

1995 (1)

C. Law, “Interaction between a moving mirror and radiation pressure: A hamiltonian formulation,” Phys. Rev. A 51(3), 2537–2541 (1995).
[Crossref]

1992 (2)

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68(5), 580–583 (1992).
[Crossref]

N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” J. Phys. A: Math. Gen. 25(21), 5677–5691 (1992).
[Crossref]

Adesso, G.

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70(2), 022318 (2004).
[Crossref]

Al-Amri, M.

W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated-emission laser,” Phys. Rev. A 88(2), 022338 (2013).
[Crossref]

W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated emission laser via cascade-driven coherence,” Phys. Rev. A 88(5), 052301 (2013).
[Crossref]

Alegre, T. M.

J. Chan, T. 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(7367), 89–92 (2011).
[Crossref]

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Allman, M.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. 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(7356), 359–363 (2011).
[Crossref]

J. Teufel, D. Li, M. Allman, K. Cicak, A. Sirois, J. Whittaker, and R. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[Crossref]

An, J.-H.

J.-H. An and W.-M. Zhang, “Non-markovian entanglement dynamics of noisy continuous-variable quantum channels,” Phys. Rev. A 76(4), 042127 (2007).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Aspelmeyer, M.

S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, “Observation of non-Markovian micromechanical Brownian motion,” Nat. Commun. 6(1), 7606 (2015).
[Crossref]

P. Sekatski, M. Aspelmeyer, and N. Sangouard, “Macroscopic optomechanics from displaced single-photon entanglement,” Phys. Rev. Lett. 112(8), 080502 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gröblacher, H. Miao, Y. Chen, M. Aspelmeyer, and O. Painter, “Laser noise in cavity-optomechanical cooling and thermometry,” New J. Phys. 15(3), 035007 (2013).
[Crossref]

J. Chan, T. 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(7367), 89–92 (2011).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. Böhm, 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(3), 030405 (2007).
[Crossref]

Bariani, F.

K. Zhang, F. Bariani, and P. Meystre, “Theory of an optomechanical quantum heat engine,” Phys. Rev. A 90(2), 023819 (2014).
[Crossref]

Bennett, S.

K. Stannigel, P. Komar, S. Habraken, S. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref]

Böhm, H.

D. Vitali, S. Gigan, A. Ferreira, H. Böhm, 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(3), 030405 (2007).
[Crossref]

Bose, S.

S. Bose, K. Jacobs, and P. L. Knight, “Scheme to probe the decoherence of a macroscopic object,” Phys. Rev. A 59(5), 3204–3210 (1999).
[Crossref]

Bouwmeester, D.

R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112(8), 080503 (2014).
[Crossref]

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91(13), 130401 (2003).
[Crossref]

Breuer, H.-P.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from markovian to non-markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

H.-P. Breuer, E.-M. Laine, and J. Piilo, “Measure for the degree of non-markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103(21), 210401 (2009).
[Crossref]

Briegel, H.-J.

W. Dür and H.-J. Briegel, “Stability of macroscopic entanglement under decoherence,” Phys. Rev. Lett. 92(18), 180403 (2004).
[Crossref]

Cai, Z.

Castin, Y.

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68(5), 580–583 (1992).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gröblacher, H. Miao, Y. Chen, M. Aspelmeyer, and O. Painter, “Laser noise in cavity-optomechanical cooling and thermometry,” New J. Phys. 15(3), 035007 (2013).
[Crossref]

J. Chan, T. 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(7367), 89–92 (2011).
[Crossref]

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
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Chen, M.

M. Chen and J. Q. You, “Non-markovian quantum state diffusion for an open quantum system in fermionic environments,” Phys. Rev. A 87(5), 052108 (2013).
[Crossref]

Chen, Y.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gröblacher, H. Miao, Y. Chen, M. Aspelmeyer, and O. Painter, “Laser noise in cavity-optomechanical cooling and thermometry,” New J. Phys. 15(3), 035007 (2013).
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Y. Chen, “Macroscopic quantum mechanics: theory and experimental concepts of optomechanics,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104001 (2013).
[Crossref]

Cheng, J.

W.-Z. Zhang, J. Cheng, W.-D. Li, and L. Zhou, “Optomechanical cooling in the non-markovian regime,” Phys. Rev. A 93(6), 063853 (2016).
[Crossref]

L. Zhou, J. Cheng, Y. Han, and W. Zhang, “Nonlinearity enhancement in optomechanical systems,” Phys. Rev. A 88(6), 063854 (2013).
[Crossref]

Churchill, L.

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. 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(7356), 359–363 (2011).
[Crossref]

J. Teufel, D. Li, M. Allman, K. Cicak, A. Sirois, J. Whittaker, and R. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[Crossref]

Cirac, J. I.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Cleuren, B.

B. Cleuren, B. Rutten, and C. Van den Broeck, “Cooling by heating: Refrigeration powered by photons,” Phys. Rev. Lett. 108(12), 120603 (2012).
[Crossref]

Cole, G. D.

S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, “Observation of non-Markovian micromechanical Brownian motion,” Nat. Commun. 6(1), 7606 (2015).
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Corn, B.

X. Zhao, J. Jing, B. Corn, and T. Yu, “Dynamics of interacting qubits coupled to a common bath: Non-markovian quantum-state-diffusion approach,” Phys. Rev. A 84(3), 032101 (2011).
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Dalibard, J.

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68(5), 580–583 (1992).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Diósi, L.

W. T. Strunz, L. Diósi, and N. Gisin, “Open system dynamics with non-markovian quantum trajectories,” Phys. Rev. Lett. 82(9), 1801–1805 (1999).
[Crossref]

T. Yu, L. Diósi, N. Gisin, and W. T. Strunz, “Non-markovian quantum-state diffusion: Perturbation approach,” Phys. Rev. A 60(1), 91–103 (1999).
[Crossref]

L. Diósi, N. Gisin, and W. T. Strunz, “Non-markovian quantum state diffusion,” Phys. Rev. A 58(3), 1699–1712 (1998).
[Crossref]

Donner, T.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. 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(7356), 359–363 (2011).
[Crossref]

Duan, L.-M.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Dür, W.

W. Dür and H.-J. Briegel, “Stability of macroscopic entanglement under decoherence,” Phys. Rev. Lett. 92(18), 180403 (2004).
[Crossref]

Eberly, J. H.

T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323(5914), 598–601 (2009).
[Crossref]

T. Yu and J. H. Eberly, “Quantum open system theory: Bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
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Eichenfield, M.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Eisert, J.

S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, “Observation of non-Markovian micromechanical Brownian motion,” Nat. Commun. 6(1), 7606 (2015).
[Crossref]

A. Mari and J. Eisert, “Cooling by heating: Very hot thermal light can significantly cool quantum systems,” Phys. Rev. Lett. 108(12), 120602 (2012).
[Crossref]

Fang, K.

Fazio, R.

F. Plastina, R. Fazio, and G. M. Palma, “Macroscopic entanglement in josephson nanocircuits,” Phys. Rev. B 64(11), 113306 (2001).
[Crossref]

Ferraro, A.

A. Ferraro, S. Olivares, and M. Paris, Gaussian States in Quantum Information, Napoli Series on physics and Astrophysics (Bibliopolis, 2005).

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. Böhm, 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(3), 030405 (2007).
[Crossref]

Ficek, Z.

Z. Ficek and R. Tanaś, “Delayed sudden birth of entanglement,” Phys. Rev. A 77(5), 054301 (2008).
[Crossref]

Flores, J. G. F.

Garraway, B. M.

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, and B. M. Garraway, “Pseudomodes as an effective description of memory: Non-markovian dynamics of two-state systems in structured reservoirs,” Phys. Rev. A 80(1), 012104 (2009).
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Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Ge, W.

W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated-emission laser,” Phys. Rev. A 88(2), 022338 (2013).
[Crossref]

W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated emission laser via cascade-driven coherence,” Phys. Rev. A 88(5), 052301 (2013).
[Crossref]

Ghobadi, R.

R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112(8), 080503 (2014).
[Crossref]

Giedke, G.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Gigan, S.

D. Vitali, S. Gigan, A. Ferreira, H. Böhm, 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(3), 030405 (2007).
[Crossref]

Giovannetti, V.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88(12), 120401 (2002).
[Crossref]

Girvin, F. M. S.

F. M. S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
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Gisin, N.

T. Yu, L. Diósi, N. Gisin, and W. T. Strunz, “Non-markovian quantum-state diffusion: Perturbation approach,” Phys. Rev. A 60(1), 91–103 (1999).
[Crossref]

W. T. Strunz, L. Diósi, and N. Gisin, “Open system dynamics with non-markovian quantum trajectories,” Phys. Rev. Lett. 82(9), 1801–1805 (1999).
[Crossref]

L. Diósi, N. Gisin, and W. T. Strunz, “Non-markovian quantum state diffusion,” Phys. Rev. A 58(3), 1699–1712 (1998).
[Crossref]

N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” J. Phys. A: Math. Gen. 25(21), 5677–5691 (1992).
[Crossref]

Gröblacher, S.

S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, “Observation of non-Markovian micromechanical Brownian motion,” Nat. Commun. 6(1), 7606 (2015).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gröblacher, H. Miao, Y. Chen, M. Aspelmeyer, and O. Painter, “Laser noise in cavity-optomechanical cooling and thermometry,” New J. Phys. 15(3), 035007 (2013).
[Crossref]

J. Chan, T. 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(7367), 89–92 (2011).
[Crossref]

Guerreiro, A.

D. Vitali, S. Gigan, A. Ferreira, H. Böhm, 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(3), 030405 (2007).
[Crossref]

Guo, G.-C.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from markovian to non-markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Habraken, S.

K. Stannigel, P. Komar, S. Habraken, S. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref]

Han, Y.

L. Zhou, J. Cheng, Y. Han, and W. Zhang, “Nonlinearity enhancement in optomechanical systems,” Phys. Rev. A 88(6), 063854 (2013).
[Crossref]

Harlow, J. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. 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(7356), 359–363 (2011).
[Crossref]

Hill, J. T.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gröblacher, H. Miao, Y. Chen, M. Aspelmeyer, and O. Painter, “Laser noise in cavity-optomechanical cooling and thermometry,” New J. Phys. 15(3), 035007 (2013).
[Crossref]

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

J. Chan, T. 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(7367), 89–92 (2011).
[Crossref]

Huang, Y.

Huang, Y.-F.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from markovian to non-markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Huelga, S. F.

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quantum non-markovianity: characterization, quantification and detection,” Rep. Prog. Phys. 77(9), 094001 (2014).
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A. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-markovianity of quantum evolutions,” Phys. Rev. Lett. 105(5), 050403 (2010).
[Crossref]

Illuminati, F.

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70(2), 022318 (2004).
[Crossref]

Jacobs, K.

S. Bose, K. Jacobs, and P. L. Knight, “Scheme to probe the decoherence of a macroscopic object,” Phys. Rev. A 59(5), 3204–3210 (1999).
[Crossref]

Jensen, K.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

Jing, J.

X. Zhao, J. Jing, J. You, and T. Yu, “Dynamics of coupled cavity arrays embedded in a non-markovian bath,” Quantum Inf. & Comput. 14, 741–756 (2014).

J. Xu, X. Zhao, J. Jing, L.-A. Wu, and T. Yu, “Perturbation methods for the non-markovian quantum state diffusion equation,” J. Phys. A: Math. Theor. 47(43), 435301 (2014).
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J. Jing, X. Zhao, J. Q. You, W. T. Strunz, and T. Yu, “Many-body quantum trajectories of non-markovian open systems,” Phys. Rev. A 88(5), 052122 (2013).
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Other (2)

W. H. Zurek, “Decoherence and the transition from quantum to classical–revisited,” arXiv preprint quant-ph/0306072 (2003).

A. Ferraro, S. Olivares, and M. Paris, Gaussian States in Quantum Information, Napoli Series on physics and Astrophysics (Bibliopolis, 2005).

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Figures (5)

Fig. 1.
Fig. 1. Sketch of the model. An F-P cavity with two movable mirrors is considered. The cavity field and two mirrors are coupled due to the radiation pressure.
Fig. 2.
Fig. 2. Maximum entanglement generation for different correlation time. The inset at the upper-right corner is the real-time entanglement dynamics, while the main plot is the maximum entanglement generation (during the entire evolution) as a function of $\gamma$. The initial state is chosen as a separable state $|1\rangle |1\rangle$. The parameters are chosen as $\omega _{1}=\omega _{2}=\omega =1$, $G_{1}=G_{2}=1$, $\Omega =\frac {2}{3}\pi$, $\kappa _{1}=\kappa _{2}=1$.
Fig. 3.
Fig. 3. Maximum entanglement generation in parametric space. The position of the red circle makers indicate the maximum entanglement is achieved for the given set of parameters $\omega$ and $\Omega$. For example, the last marker shows that when $\omega /G=4$, the maximum entanglement appears at $\Omega /G\approx 8$. In the plot, we choose the symmetric case $\omega _{1}=\omega _{2}=\omega$, $G_{1}=G_{2}=G$, $\kappa _{1}=\kappa _{2}=1$, and $\gamma =0.4$.
Fig. 4.
Fig. 4. Entanglement dynamics for initial state $(c_{1}|1\rangle +c_{2}|-1\rangle )\otimes |1\rangle$. The color reflects the value of the variable of $E_{N}$ which is an indicator of entanglement. The parameters are chosen as $\omega _{1}=\omega _{2}=\omega =1$, $G_{1}=G_{2}=1$, $\gamma =0.4$, $\Omega =\pi /3$, $\kappa _{1}=\kappa _{2}=1$.
Fig. 5.
Fig. 5. Entanglement dynamics ($E_{N}$) for different memory time. The initial state is chosen as the “cat state” $|\psi _{ini}\rangle \propto (|1\rangle |1\rangle +|-1\rangle |-1\rangle )$. The parameters are chosen as $\omega _{1}=\omega _{2}=\omega =1$, $G_{1}=G_{2}=1$, $\Omega =0$, $\kappa _{1}=\kappa _{2}=1$.

Equations (53)

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H = H S + H B + H i n t ,
H S = ω 1 ( p 1 2 + q 1 2 ) + ω 2 ( p 2 2 + q 2 2 ) + ω c a a + G 1 a a q 1 + G 2 a a q 2 ,
H B = i ν i b i b i ,
H i n t = i g i ( κ 1 q 1 + κ 2 q 2 ) ( b i + b i ) .
t | ψ t ( z ) = [ i H s + L z t L 0 t d s α ( t , s ) δ δ z s ] | ψ t ( z ) ,
t | ψ t ( z ) = [ i H s + L z t L O ¯ ] | ψ t ( z ) ,
O ( t , s , z ) = i = 1 5 f i ( t , s ) O i + 0 t d s f 6 ( t , s , s ) z s ,
O 1 = q 1 , O 2 = q 2 , O 3 = p 1 , O 4 = p 2 , O 5 = a a ,
t f 1 = 2 ω 1 f 3 2 i κ 1 F 1 f 3 i κ 1 F 2 f 4 + i κ 1 F 3 f 1 + i κ 1 F 4 f 2 i κ 2 F 1 f 4 κ 1 F 6 ,
t f 2 = 2 ω 2 f 4 i κ 1 F 2 f 3 i κ 2 F 1 f 3 2 i κ 2 F 1 f 4 + i κ 2 F 3 f 1 + i κ 2 F 4 f 2 κ 2 F 6 ,
t f 3 = 2 ω 1 f 1 i κ 1 F 3 f 3 i κ 2 F 3 f 4 ,
t f 4 = 2 ω 2 f 2 i κ 1 F 4 f 3 i κ 2 F 4 f 4 ,
t f 5 = G 1 f 3 + G 2 f 4 i κ 1 F 5 f 3 i κ 2 F 5 f 4 ,
t f 6 ( t , s , s ) = i κ 1 f 3 ( t , s ) F 6 ( t , s ) i κ 2 f 4 ( t , s ) F 6 ( t , s ) ,
f 1 ( t , t ) = κ 1 ,
f 2 ( t , t ) = κ 2 ,
f 3 ( t , t ) = f 4 ( t , t ) = f 5 ( t , t ) = 0 ,
f 6 ( t , t , s ) = 0 , f 6 ( t , s , t ) = i ( κ 1 + κ 2 ) .
d d t ρ = i ω 1 ( p 1 p 1 ρ ρ p 1 p 1 ) i ω 1 ( q 1 q 1 ρ ρ q 1 q 1 ) i ω 2 ( p 2 p 2 ρ ρ p 2 p 2 ) i ω 2 ( q 2 q 2 ρ ρ q 2 q 2 ) i G 1 ( q 1 a a ρ ρ q 1 a a ) i G 2 ( q 2 a a ρ ρ q 2 a a ) + κ 1 F 1 ( q 1 ρ q 1 ρ q 1 q 1 ) + κ 1 F 1 ( q 1 ρ q 1 q 1 q 1 ρ ) + κ 1 F 2 ( q 1 ρ q 2 ρ q 2 q 1 ) + κ 1 F 2 ( q 2 ρ q 1 q 1 q 2 ρ ) + κ 1 F 3 ( q 1 ρ p 1 ρ p 1 q 1 ) + κ 1 F 3 ( p 1 ρ q 1 q 1 p 1 ρ ) + κ 1 F 4 ( q 1 ρ p 2 ρ p 2 q 1 ) + κ 1 F 4 ( p 2 ρ q 1 q 1 p 2 ρ ) + κ 1 F 5 ( q 1 ρ a a ρ a a q 1 ) + κ 1 F 5 ( a a ρ q 1 q 1 a a ρ ) + κ 2 F 1 ( q 2 ρ q 1 ρ q 1 q 2 ) + κ 2 F 1 ( q 1 ρ q 2 q 2 q 1 ρ ) + κ 2 F 2 ( q 2 ρ q 2 ρ q 2 q 2 ) + κ 2 F 2 ( q 2 ρ q 2 q 2 q 2 ρ ) + κ 2 F 3 ( q 2 ρ p 1 ρ p 1 q 2 ) + κ 2 F 3 ( p 1 ρ q 2 q 2 p 1 ρ ) + κ 2 F 4 ( q 2 ρ p 2 ρ p 2 q 2 ) + κ 2 F 4 ( p 2 ρ q 2 q 2 p 2 ρ ) + κ 2 F 5 ( q 2 ρ a a ρ q 2 a a ) + κ 2 F 5 ( a a ρ q 2 q 2 a a ρ ) .
E N ( ρ ) = max [ 0 , ln ν ] ,
α = γ 2 e ( γ + i Ω ) | t s | .
J ( ω ) = γ / 2 π ( ω Ω ) 2 + γ 2 ,
| ψ i n i = ( c 1 | 1 + c 2 | 1 ) | 1 ,
H B = m ω m b 1 , m b 1 , m + n ν n b 2 , n b 2 , n .
H i n t = m g 1 , m L 1 b 1 , m + m g 2 , n L 2 b 2 , n + H . c . .
t | ψ ( t , z 1 , z 2 ) = [ i H S + L 1 z 1 , t L 1 0 t d s α 1 ( t , s ) δ δ z 1 , s + L 2 z 2 , t L 2 0 t d s α 2 ( t , s ) δ δ z 2 , s ] | ψ ( t , z 1 , z 2 ) ,
z 1 , t = m g 1 , m z 1 , m e i ω m t ,
z 2 , t = n g 2 , n z 2 , n e i ν n t .
α 1 ( t , s ) = m | g 1 , m | 2 e i ω m ( t s ) ,
α 2 ( t , s ) = n | g 2 , n | 2 e i ν n ( t s ) .
t | ψ ( t , z , w ) = [ i H S + L z t + L w t L O ¯ 1 L O ¯ 2 ] | ψ ( t , z , w )
t O 1 = [ i H s + L z t + L w t L O ¯ 1 L O ¯ 2 , O 1 ] L δ δ z s O ¯ 1 L δ δ z s O ¯ 2 ,
t O 2 = [ i H s + L z t + L w t L O ¯ 1 L O ¯ 2 , O 2 ] L δ δ w s O ¯ 1 L δ δ w s O ¯ 2 ,
d d t q 1 = 2 ω 1 p 1 ,
d d t q 2 = 2 ω 2 p 2 ,
d d t p 1 = 2 ω 1 q 1 G 1 a a + i i = 1 5 ( κ 1 F i O i + κ 1 F i O i ) ,
d d t p 2 = 2 ω 2 q 2 G 2 a a + i i = 1 5 ( κ 2 F i O i + κ 2 F i O i ) ,
d d t a a = 0 ,
d d t p 1 p 1 = i ω 1 ( 2 4 i p 1 q 1 ) 2 G 1 p 1 a a + 2 i i = 1 5 ( κ 1 F i p 1 O i κ 1 F i O i p 1 ) ,
d d t p 1 q 1 = 2 ω 1 p 1 p 1 2 ω 1 q 1 q 1 G 1 q 1 a a + i i = 1 5 ( κ 1 F i q 1 O i κ 1 F i O i q 1 ) ,
d d t p 1 p 2 = 2 ω 1 q 1 p 2 2 ω 2 p 1 q 2 G 1 p 2 a a G 2 p 1 a a + i i = 1 5 ( κ 1 F i p 2 O i κ 1 F i O i p 2 ) + i i = 1 5 ( κ 2 F i p 1 O i κ 2 F i O i p 1 ) ,
d d t p 1 q 2 = 2 ω 1 q 1 q 2 + 2 ω 2 p 1 p 2 G 1 q 2 a a + i i = 1 5 ( κ 1 F i q 2 O i κ 1 F i O i q 2 ) ,
d d t q 1 q 1 = i ω 1 ( 2 + 4 i p 1 q 1 ) ,
d d t q 1 p 2 = 2 ω 1 p 1 p 2 2 ω 2 q 1 q 2 G 2 q 1 a a + i i = 1 5 ( κ 2 F i q 1 O i κ 2 F i O i q 1 ) ,
d d t q 1 q 2 = 2 ω 1 p 1 q 2 + 2 ω 2 q 1 p 2 ,
d d t p 2 p 2 = 2 i i = 1 5 ( κ 2 F i p 2 O i κ 2 F i O i p 2 ) i ω 2 ( 2 4 i p 2 q 2 ) 2 G 2 p 2 a a ,
d d t p 2 q 2 = 2 ω 2 p 2 p 2 2 ω 2 q 2 q 2 G 2 q 2 a a + i i = 1 5 ( κ 2 F i q 2 O i κ 2 F i O i q 2 ) ,
d d t q 2 q 2 = i ω 2 ( 2 + 4 i p 2 q 2 ) ,
d d t q 1 a a = 2 ω 1 p 1 a a ,
d d t q 2 a a = 2 ω 2 p 2 a a ,
d d t p 1 a a = i = 1 5 i κ 1 F i O i a a + i κ 1 F i O i a a 2 ω 1 q 1 a a G 1 a a a a ,
d d t p 2 a a = i = 1 5 i κ 2 F i O i a a + i κ 2 F i O i a a 2 ω 2 q 2 a a G 2 a a a a ,
d d t a a a a = 0.

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