Abstract

In this paper, we propose a new theoretical scheme for generating a macroscopic Schrödinger cat state of a mechanical oscillator in a hybrid optomechanical system where a beam of two-level atoms passes through the cavity. In the model under consideration, the cavity field couples to the macroscopic mirror through the optomechanical interaction while it couples to the atom through a generalized Jaynes–Cummings interaction that involves the cavity-mode structure. The motion of the mirror modifies the cavity-mode function and therefore modulates the atom-field interaction, leading to the three-mode atom-field-mirror coupling or, equivalently, polariton-mirror coupling in a dressed picture. This interaction induces a controllable anharmonicity in the energy spectrum of the mechanical oscillator, which provides the possibility of generating a superposition of two time-dependent coherent states of the mechanical oscillator just by performing a conditional measurement on the internal states of the atoms exiting the optomechanical cavity. We also investigate the tripartite atom-field-mirror entanglement, which is controllable by adjusting the parameters of the system. In addition, we explore the effects of the mechanical dissipation and thermal noise on the tripartite quantum correlation in the system as well as the generated mechanical superposition state.

© 2020 Optical Society of America

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2020 (2)

J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas, “Optomechanics with levitated particles,” Rep. Prog. Phys. 83, 026401 (2020).
[Crossref]

Z. Yang, S. L. Chao, and L. Zhou, “Generating macroscopic quantum superposition and a phonon laser in a hybrid optomechanical system,” J. Opt. Soc. Am. B 37, 1–8 (2020).
[Crossref]

2019 (5)

H. Xie, X. Shang, C.-G. Liao, Z.-H. Chen, and X.-M. Lin, “Macroscopic superposition states of a mechanical oscillator in an optomechanical system with quadratic coupling,” Phys. Rev. A 100, 033803 (2019).
[Crossref]

V. Bergholm, W. Wieczorek, T. Schulte-Herbrüggen, and M. Keyl, “Optimal control of hybrid optomechanical systems for generating non-classical states of mechanical motion,” Quantum Sci. Technol. 4, 034001 (2019).
[Crossref]

O. Černotík, C. Genes, and A. Dantan, “Interference effects in hybrid cavity optomechanics,” Quantum Sci. Technol. 4, 024002 (2019).
[Crossref]

O. Černotík, A. Dantan, and C. Genes, “Cavity quantum electrodynamics with frequency-dependent reflectors,” Phys. Rev. Lett. 122, 243601 (2019).
[Crossref]

M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht, “Testing the gravitational field generated by a quantum superposition,” New J. Phys. 21, 093052 (2019).
[Crossref]

2018 (5)

C. Eichler and J. R. Petta, “Realizing a circuit analog of an optomechanical system with longitudinally coupled superconducting resonators,” Phys. Rev. Lett. 120, 227702 (2018).
[Crossref]

R. Y. Teh, S. Kiesewetter, P. D. Drummond, and M. D. Reid, “Creation, storage, and retrieval of an optomechanical cat state,” Phys. Rev. A 98, 063814 (2018).
[Crossref]

S. Asiri, Z. Liao, and M. S. Zubairy, “Quantum-state reconstruction of a mechanical mirror in a hybrid system,” Phys. Rev. A 98, 043815 (2018).
[Crossref]

M. Ringbauer, T. J. Weinhold, L. Howard, A. White, and M. Vanner, “Generation of mechanical interference fringes by multi-photon counting,” New J. Phys. 20, 053042 (2018).
[Crossref]

J. Clarke and M. R. Vanner, “Growing macroscopic superposition states via cavity quantum optomechanics,” Quantum Sci. Technol. 4, 014003 (2018).
[Crossref]

2017 (4)

M. Cotrufo, A. Fiore, and E. Verhagen, “Coherent atom-phonon interaction through mode field coupling in hybrid optomechanical systems,” Phys. Rev. Lett. 118, 133603 (2017).
[Crossref]

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, 191–195 (2017).
[Crossref]

A. Dalafi and M. H. Naderi, “Controlling steady-state bipartite entanglement and quadrature squeezing in a membrane-in-the-middle optomechanical system with two Bose-Einstein condensates,”Phys. Rev. A 96, 033631 (2017).
[Crossref]

J. H. Liu, Y. B. Zhang, Y.-F. Yu, and Z.-M. Zhang, “Entangling cavity optomechanical systems via a flying atom,” Opt. Express 25,7592–7603 (2017).
[Crossref]

2016 (6)

S. V. Kusminskiy, H. X. Tang, and F. Marquardt, “Coupled spin-light dynamics in cavity optomagnonics,” Phys. Rev. A 94, 033821 (2016).
[Crossref]

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref]

J.-Q. Liao, J.-F. Huang, and L. Tian, “Generation of macroscopic Schrödinger-cat states in qubit-oscillator systems,” Phys. Rev. A 93, 033853 (2016).
[Crossref]

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116, 163602 (2016).
[Crossref]

T. Milburn, M. Kim, and M. Vanner, “Nonclassical-state generation in macroscopic systems via hybrid discrete-continuous quantum measurements,” Phys. Rev. A 93, 053818 (2016).
[Crossref]

U. B. Hoff, J. Kollath-Bönig, J. S. Neergaard-Nielsen, and U. L. Andersen, “Measurement-induced macroscopic superposition states in cavity optomechanics,” Phys. Rev. Lett. 117, 143601 (2016).
[Crossref]

2015 (7)

A. Carlisle, H. Kwon, H. Jeong, A. Ferraro, and M. Paternostro, “Limitations of a measurement-assisted optomechanical route to quantum macroscopicity of superposition states,” Phys. Rev. A 92, 022123 (2015).
[Crossref]

W. Ge and M. S. Zubairy, “Macroscopic optomechanical superposition via periodic qubit flipping,” Phys. Rev. A 91, 013842 (2015).
[Crossref]

E. E. Wollman, C. Lei, A. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. Schwab, “Quantum squeezing of motion in a mechanical resonator,” Science 349, 952–955 (2015).
[Crossref]

F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Quantum nondemolition measurement of a nonclassical state of a massive object,” Phys. Rev. X 5, 041037 (2015).
[Crossref]

J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115, 243601 (2015).
[Crossref]

J. M. Pirkkalainen, S. Cho, F. Massel, J. Tuorila, T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
[Crossref]

A. Jöckel, A. Faber, T. Kampschulte, M. Korppi, M. T. Rakher, and P. Treutlein, “Sympathetic cooling of a membrane oscillator in a hybrid mechanical–atomic system,” Nat. Nanotechnol. 10, 55 (2015).
[Crossref]

2014 (4)

J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014).
[Crossref]

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2, 11–43 (2014).
[Crossref]

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

M. Arndt and K. Hornberger, “Testing the limits of quantum mechanical superpositions,” Nat. Phys. 10, 271–277 (2014).
[Crossref]

2013 (5)

T. Palomaki, J. Teufel, R. Simmonds, and K. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref]

T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett. 110, 193602 (2013).
[Crossref]

M. Blencowe, “Effective field theory approach to gravitationally induced decoherence,” Phys. Rev. Lett. 111, 021302 (2013).
[Crossref]

H. Tan, F. Bariani, G. Li, and P. Meystre, “Generation of macroscopic quantum superpositions of optomechanical oscillatos by dissipation,” Phys. Rev. A 88, 023817 (2013).
[Crossref]

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
[Crossref]

2012 (3)

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

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109, 063601 (2012).
[Crossref]

B. Rogers, M. Paternostro, G. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,”Phys. Rev. A 86, 042323 (2012).
[Crossref]

2011 (5)

A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Jezek, and U. L. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A 84, 050301 (2011).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011).
[Crossref]

C. Regal and K. Lehnert, “From cavity electromechanics to cavity optomechanics,” J. Phys. Conf. Ser. 264, 012025 (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, 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, 89–92 (2011).
[Crossref]

2010 (5)

N. Brahms and D. M. Stamper-Kurn, “Spin optodynamics analog of cavity optomechanics,” Phys. Rev. A 82, 041804 (2010).
[Crossref]

P. Marek and J. Fiurasek, “Elementary gates for quantum information with superposed coherent states,” Phys. Rev. A 82, 014304 (2010).
[Crossref]

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010).
[Crossref]

U. Akram, N. Kiesel, M. Aspelmeyer, and G. J. Milburn, “Single-photon opto-mechanics in the strong coupling regime,” New J. Phys. 12, 083030 (2010).
[Crossref]

F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, and Y. Chen, “Preparing a mechanical oscillator in non-Gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
[Crossref]

2009 (1)

K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. 102, 020501 (2009).
[Crossref]

2008 (3)

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref]

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

C. Sabín and G. García-Alcaine, “A classification of entanglement in three-qubit systems,” Eur. Phys. J. D 48, 435–442 (2008).
[Crossref]

2006 (2)

G. Khitrova, H. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nat. Phys. 2, 81–90 (2006).
[Crossref]

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

2004 (1)

S. Rinner, H. Walther, and E. Werner, “How to measure the decoherence of a micromaser field under well controlled conditions,”Phys. Rev. Lett. 93, 160407 (2004).
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2003 (2)

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

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
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2001 (2)

X. Wang, “Quantum teleportation of entangled coherent states,” Phys. Rev. A 64, 022302 (2001).
[Crossref]

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565 (2001).
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1997 (1)

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

1996 (1)

M. F. Bocko and R. Onofrio, “On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress,”Rev. Mod. Phys. 68, 755 (1996).
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1995 (1)

V. Buzek and P. L. Knight, “I: quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 34, 1–158 (1995).
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1992 (1)

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, and K. S. Thorne, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
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Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, and K. S. Thorne, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
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Akram, U.

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
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U. Akram, N. Kiesel, M. Aspelmeyer, and G. J. Milburn, “Single-photon opto-mechanics in the strong coupling regime,” New J. Phys. 12, 083030 (2010).
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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, 89–92 (2011).
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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, 359–363 (2011).
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Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, and K. S. Thorne, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
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Anant, V.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
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Andersen, U. L.

U. B. Hoff, J. Kollath-Bönig, J. S. Neergaard-Nielsen, and U. L. Andersen, “Measurement-induced macroscopic superposition states in cavity optomechanics,” Phys. Rev. Lett. 117, 143601 (2016).
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A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Jezek, and U. L. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A 84, 050301 (2011).
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Appel, J.

R. A. Thomas, M. Parniak, C. Østfeldt, C. B. Møller, C. Bærentsen, Y. Tsaturyan, A. Schliesser, J. Appel, E. Zeuthen, and E. S. Polzik, “Entanglement between distant macroscopic mechanical and spin systems,” arXiv:2003.11310 (2020).

Arcizet, O.

F. Fogliano, B. Besga, A. Reigue, P. Heringlake, L. M. de Lépinay, C. Vaneph, J. Reichel, B. Pigeau, and O. Arcizet, “Cavity nano-optomechanics in the ultrastrong coupling regime with ultrasensitive force sensors,” arXiv:1904.01140 (2019).

Arndt, M.

M. Arndt and K. Hornberger, “Testing the limits of quantum mechanical superpositions,” Nat. Phys. 10, 271–277 (2014).
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S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010).
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S. Asiri, Z. Liao, and M. S. Zubairy, “Quantum-state reconstruction of a mechanical mirror in a hybrid system,” Phys. Rev. A 98, 043815 (2018).
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G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).

Aspelmeyer, M.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
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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, 89–92 (2011).
[Crossref]

U. Akram, N. Kiesel, M. Aspelmeyer, and G. J. Milburn, “Single-photon opto-mechanics in the strong coupling regime,” New J. Phys. 12, 083030 (2010).
[Crossref]

K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. 102, 020501 (2009).
[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 541, 191–195 (2017).
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F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Quantum nondemolition measurement of a nonclassical state of a massive object,” Phys. Rev. X 5, 041037 (2015).
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Bærentsen, C.

R. A. Thomas, M. Parniak, C. Østfeldt, C. B. Møller, C. Bærentsen, Y. Tsaturyan, A. Schliesser, J. Appel, E. Zeuthen, and E. S. Polzik, “Entanglement between distant macroscopic mechanical and spin systems,” arXiv:2003.11310 (2020).

Bariani, F.

H. Tan, F. Bariani, G. Li, and P. Meystre, “Generation of macroscopic quantum superpositions of optomechanical oscillatos by dissipation,” Phys. Rev. A 88, 023817 (2013).
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Bassi, A.

M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht, “Testing the gravitational field generated by a quantum superposition,” New J. Phys. 21, 093052 (2019).
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Becker, T.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
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Bergholm, V.

V. Bergholm, W. Wieczorek, T. Schulte-Herbrüggen, and M. Keyl, “Optimal control of hybrid optomechanical systems for generating non-classical states of mechanical motion,” Quantum Sci. Technol. 4, 034001 (2019).
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Besga, B.

F. Fogliano, B. Besga, A. Reigue, P. Heringlake, L. M. de Lépinay, C. Vaneph, J. Reichel, B. Pigeau, and O. Arcizet, “Cavity nano-optomechanics in the ultrastrong coupling regime with ultrasensitive force sensors,” arXiv:1904.01140 (2019).

Blencowe, M.

M. Blencowe, “Effective field theory approach to gravitationally induced decoherence,” Phys. Rev. Lett. 111, 021302 (2013).
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Bocko, M. F.

M. F. Bocko and R. Onofrio, “On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress,”Rev. Mod. Phys. 68, 755 (1996).
[Crossref]

Bose, S.

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

Bouwmeester, D.

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

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

Bowen, W. P.

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
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Brahms, N.

N. Brahms and D. M. Stamper-Kurn, “Spin optodynamics analog of cavity optomechanics,” Phys. Rev. A 82, 041804 (2010).
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Brandt, M.

J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115, 243601 (2015).
[Crossref]

Brennecke, F.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref]

Brune, M.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565 (2001).
[Crossref]

Buzek, V.

V. Buzek and P. L. Knight, “I: quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 34, 1–158 (1995).
[Crossref]

Carlesso, M.

M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht, “Testing the gravitational field generated by a quantum superposition,” New J. Phys. 21, 093052 (2019).
[Crossref]

Carlisle, A.

A. Carlisle, H. Kwon, H. Jeong, A. Ferraro, and M. Paternostro, “Limitations of a measurement-assisted optomechanical route to quantum macroscopicity of superposition states,” Phys. Rev. A 92, 022123 (2015).
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Cernotík, O.

O. Černotík, C. Genes, and A. Dantan, “Interference effects in hybrid cavity optomechanics,” Quantum Sci. Technol. 4, 024002 (2019).
[Crossref]

O. Černotík, A. Dantan, and C. Genes, “Cavity quantum electrodynamics with frequency-dependent reflectors,” Phys. Rev. Lett. 122, 243601 (2019).
[Crossref]

Chan, J.

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, 89–92 (2011).
[Crossref]

Chao, S. L.

Chen, Y.

F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, and Y. Chen, “Preparing a mechanical oscillator in non-Gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
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Chen, Z.-H.

H. Xie, X. Shang, C.-G. Liao, Z.-H. Chen, and X.-M. Lin, “Macroscopic superposition states of a mechanical oscillator in an optomechanical system with quadratic coupling,” Phys. Rev. A 100, 033803 (2019).
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Cho, S.

J. M. Pirkkalainen, S. Cho, F. Massel, J. Tuorila, T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
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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, 359–363 (2011).
[Crossref]

Ciuti, C.

J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014).
[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, 191–195 (2017).
[Crossref]

F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Quantum nondemolition measurement of a nonclassical state of a massive object,” Phys. Rev. X 5, 041037 (2015).
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Clarke, J.

J. Clarke and M. R. Vanner, “Growing macroscopic superposition states via cavity quantum optomechanics,” Quantum Sci. Technol. 4, 014003 (2018).
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Clerk, A. A.

E. E. Wollman, C. Lei, A. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. Schwab, “Quantum squeezing of motion in a mechanical resonator,” Science 349, 952–955 (2015).
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Cotrufo, M.

M. Cotrufo, A. Fiore, and E. Verhagen, “Coherent atom-phonon interaction through mode field coupling in hybrid optomechanical systems,” Phys. Rev. Lett. 118, 133603 (2017).
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Dalafi, A.

A. Dalafi and M. H. Naderi, “Controlling steady-state bipartite entanglement and quadrature squeezing in a membrane-in-the-middle optomechanical system with two Bose-Einstein condensates,”Phys. Rev. A 96, 033631 (2017).
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Damskägg, E.

J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115, 243601 (2015).
[Crossref]

Danilishin, S.

F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, and Y. Chen, “Preparing a mechanical oscillator in non-Gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
[Crossref]

Dantan, A.

O. Černotík, C. Genes, and A. Dantan, “Interference effects in hybrid cavity optomechanics,” Quantum Sci. Technol. 4, 024002 (2019).
[Crossref]

O. Černotík, A. Dantan, and C. Genes, “Cavity quantum electrodynamics with frequency-dependent reflectors,” Phys. Rev. Lett. 122, 243601 (2019).
[Crossref]

De Chiara, G.

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2, 11–43 (2014).
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B. Rogers, M. Paternostro, G. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,”Phys. Rev. A 86, 042323 (2012).
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F. Fogliano, B. Besga, A. Reigue, P. Heringlake, L. M. de Lépinay, C. Vaneph, J. Reichel, B. Pigeau, and O. Arcizet, “Cavity nano-optomechanics in the ultrastrong coupling regime with ultrasensitive force sensors,” arXiv:1904.01140 (2019).

Devoret, M. H.

A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, “The Kerr-cat qubit: stabilization, readout, and gates,” arXiv:1907.12131 (2019).

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]

Dong, R.

A. Tipsmark, R. Dong, A. Laghaout, P. Marek, M. Jezek, and U. L. Andersen, “Experimental demonstration of a Hadamard gate for coherent state qubits,” Phys. Rev. A 84, 050301 (2011).
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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, 359–363 (2011).
[Crossref]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref]

Drever, R. W.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, and K. S. Thorne, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
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R. Y. Teh, S. Kiesewetter, P. D. Drummond, and M. D. Reid, “Creation, storage, and retrieval of an optomechanical cat state,” Phys. Rev. A 98, 063814 (2018).
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C. Eichler and J. R. Petta, “Realizing a circuit analog of an optomechanical system with longitudinally coupled superconducting resonators,” Phys. Rev. Lett. 120, 227702 (2018).
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Englert, B.-G.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Esslinger, T.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref]

Faber, A.

A. Jöckel, A. Faber, T. Kampschulte, M. Korppi, M. T. Rakher, and P. Treutlein, “Sympathetic cooling of a membrane oscillator in a hybrid mechanical–atomic system,” Nat. Nanotechnol. 10, 55 (2015).
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Fabre, C.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).

Favero, I.

J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014).
[Crossref]

Ferraro, A.

A. Carlisle, H. Kwon, H. Jeong, A. Ferraro, and M. Paternostro, “Limitations of a measurement-assisted optomechanical route to quantum macroscopicity of superposition states,” Phys. Rev. A 92, 022123 (2015).
[Crossref]

Fiore, A.

M. Cotrufo, A. Fiore, and E. Verhagen, “Coherent atom-phonon interaction through mode field coupling in hybrid optomechanical systems,” Phys. Rev. Lett. 118, 133603 (2017).
[Crossref]

Fiurasek, J.

P. Marek and J. Fiurasek, “Elementary gates for quantum information with superposed coherent states,” Phys. Rev. A 82, 014304 (2010).
[Crossref]

Fogliano, F.

F. Fogliano, B. Besga, A. Reigue, P. Heringlake, L. M. de Lépinay, C. Vaneph, J. Reichel, B. Pigeau, and O. Arcizet, “Cavity nano-optomechanics in the ultrastrong coupling regime with ultrasensitive force sensors,” arXiv:1904.01140 (2019).

Frattini, N. E.

A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, “The Kerr-cat qubit: stabilization, readout, and gates,” arXiv:1907.12131 (2019).

García-Alcaine, G.

C. Sabín and G. García-Alcaine, “A classification of entanglement in three-qubit systems,” Eur. Phys. J. D 48, 435–442 (2008).
[Crossref]

Gardiner, C.

C. Gardiner and P. Zoller, Quantum Noise: a Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, 2004).

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W. Ge and M. S. Zubairy, “Macroscopic optomechanical superposition via periodic qubit flipping,” Phys. Rev. A 91, 013842 (2015).
[Crossref]

Genes, C.

O. Černotík, A. Dantan, and C. Genes, “Cavity quantum electrodynamics with frequency-dependent reflectors,” Phys. Rev. Lett. 122, 243601 (2019).
[Crossref]

O. Černotík, C. Genes, and A. Dantan, “Interference effects in hybrid cavity optomechanics,” Quantum Sci. Technol. 4, 024002 (2019).
[Crossref]

Ghobadi, R.

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

Gibbs, H.

G. Khitrova, H. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nat. Phys. 2, 81–90 (2006).
[Crossref]

Girvin, S. M.

A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, “The Kerr-cat qubit: stabilization, readout, and gates,” arXiv:1907.12131 (2019).

Grimm, A.

A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, “The Kerr-cat qubit: stabilization, readout, and gates,” arXiv:1907.12131 (2019).

Gröblacher, S.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[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, 89–92 (2011).
[Crossref]

Grynberg, G.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light (Cambridge University, 2010).

Gullo, N. L.

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2, 11–43 (2014).
[Crossref]

Gürsel, Y.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, and K. S. Thorne, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[Crossref]

Hakonen, P.

J. M. Pirkkalainen, S. Cho, F. Massel, J. Tuorila, T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
[Crossref]

Hammerer, K.

K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. 102, 020501 (2009).
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Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of an optomechanical cavity with optical frequency $ {\omega _c} $, mechanical frequency $ {\omega _m} $, and a flying two-level atom with transition frequency $ {\omega _a} $ passing through the cavity along the $x$ direction. The atom is directly coupled to the cavity field described by a generalized Jaynes–Cummings model, where the atom-field coupling depends on the mode structure of the cavity field, while it is indirectly coupled to mechanical oscillator due to the modification of the spatial distribution of the cavity field.
Fig. 2.
Fig. 2. (a) Anharmonicity of the energy spectrum of the MO for a fixed phonon number $m$ and different photon numbers $n = 0,1,2$. (b) The normalized potential of the MO and (c)–(f) its modified form for the normalized optomechanical coupling parameter $\xi /{\omega _m} = 0.5$ and for different values of the normalized atom-field-mirror coupling parameter as well as the intracavity photon number: (c) $\lambda /{\omega _m} = 0$, $n = 1$; (d) $\lambda /{\omega _m} = 0$, $n = 10$; (e) $\lambda /{\omega _m} = 0.25$, $n = 1$; and (f) $\lambda /{\omega _m} = 0.25$, $n = 10$. The black dots in each panel indicate the location of the minima of the mechanical oscillator potential.
Fig. 3.
Fig. 3. Tripartite atom-field-mirror negativity as a function of ${\omega _m}t$. The left and right panels show, respectively, the atom-field-mirror negativity for $\xi /{\omega _m} = 0.1$ and $\xi /{\omega _m} = 0.5$, and for different values of coupling ratio $\lambda /\xi$: (a), (d) $\lambda /\xi = 0.01$ [(a) $\lambda /{\omega _m} = 0.001$, (d) $\lambda /{\omega _m} = 0.005$], (b), (e) $\lambda /\xi = 0.1$ [(b) $\lambda /{\omega _m} = 0.01$, (e) $\lambda /{\omega _m} = 0.05$], and (c), (f) $\lambda /\xi = 0.5$ [(c) $\lambda /{\omega _m} = 0.05$, (f) $\lambda /{\omega _m} = 0.25$]. The blue solid and red dashed lines in each panel show the tripartite entanglement for photon numbers $n = 1$ and $n = 10$, respectively.
Fig. 4.
Fig. 4. (a) and (d) Contour plots of the time evolution of the Wigner distributions $W_m^{(\pm)}$ corresponding to (a) the state ${| {{\chi _ +}(t)} \rangle _m}$ and (d) the state ${| {{\chi _ -}(t)} \rangle _m}$ for $\xi /{\omega _m} = 0.5$, $\lambda /{\omega _m} = 0.25$, and $n = 10$. Panels (b), (c) and (e), (f) show, respectively, $W_m^{(+)}$ and $W_m^{(-)}$ at time $t = \pi /{\omega _m}$ but for different values of photon number: (b), (e) $n = 10$ and (c), (f) $n = 1$.
Fig. 5.
Fig. 5. Tripartite atom-field-mirror negativity as a function of the scaled time ${\omega _m}t$ for different values of the normalized decoherence rate ${\Gamma _m}/{\omega _m} = {\gamma _m}{n_{\rm th}}/{\omega _m}$ and two values of the intracavity photon number $n = 1$ [(a),(b)] and $n = 10$ [(c),(d)]. In (a) and (c) we have set ${\Gamma _m}/{\omega _m} = 0$ (${n_{\rm th}} = 0$) and ${\gamma _m}/{\omega _m}{= 10^{- 6}}$ [blue solid line], ${\gamma _m}/{\omega _m}{= 10^{- 3}}$ [red dashed line], ${\gamma _m}/{\omega _m}{= 10^{- 2}}$ [green dot-dashed line], and ${\gamma _m}/{\omega _m} = 0.1$ [black dotted line]. In (b) and (d) we have set ${\Gamma _m}/{\omega _m} \ne 0$: ${\Gamma _m}/{\omega _m} = 0.01$ (${\gamma _m}/{\omega _m}{= 10^{- 6}}$, ${n_{\rm th}}{= 10^4}$) [blue thick line], ${\Gamma _m}/{\omega _m} = 0.1$ (${\gamma _m}/{\omega _m}{= 10^{- 6}}$, ${n_{\rm th}}{= 10^5}$) [red dashed line], and ${\Gamma _m}/{\omega _m} = 0.3$ (${\gamma _m}/{\omega _m}{= 10^{- 6}}$, ${n_{\rm th}} = 3 \times {10^5}$) [green dot-dashed line]. Here the normalized optomechanical coupling parameter $\xi /{\omega _m}$ and the tripartite coupling parameter $\lambda /{\omega _m}$ are fixed to be $\xi /{\omega _m} = 0.5$ and $\lambda /{\omega _m} = 0.25$.
Fig. 6.
Fig. 6. 3D Plots of the Wigner distributions corresponding to the states $\hat \rho _m^{(+)}(t)$ [left panels] and $\hat \rho _m^{(-)}(t)$ [right panels] with the intracavity photon number $n = 10$ at $t = \pi /{\omega _m}$ in the presence of mechanical dissipation and thermal noise for constant value of ${\gamma _m}/{\omega _m}{= 10^{- 6}}$ but different decoherence rates (a),(c) ${\Gamma _m}/{\omega _m} = 0.01$ (${n_{\rm th}}{= 10^4}$) and (b),(d) ${\Gamma _m}/{\omega _m} = 0.1$ (${n_{\rm th}}{= 10^5}$). The other parameters are the same as those in Fig. 4.

Equations (43)

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H ^ = H ^ o p t + H ^ a t o m ,
H ^ o p t = ω c a ^ a ^ + ω m b ^ b ^ ξ a ^ a ^ ( b ^ + b ^ )
H ^ a t o m = P ^ 2 2 M + ω a 2 σ ^ z + g ^ ( x , y , z ) ( a ^ σ ^ + + a ^ σ ^ ) ,
g ^ = g ^ ( x , z ) = g 0 sin ( k x x ) ( sin ( k z z ) q ^ l z k z z cos ( k z z ) ) .
H ^ = ω c a ^ a ^ + ω m b ^ b ^ + ω a 2 σ ^ z ξ a ^ a ^ ( b ^ + b ^ ) + [ G 0 λ ( b ^ + b ^ ) ] ( a ^ σ ^ + + a ^ σ ^ ) ,
G 0 = g 0 sin ( k x x ) sin ( k z z ) ,
λ = 2 m o ω m g 0 l z k z z sin ( k x x ) cos ( k z z ) .
H ^ = H ~ ^ + ω m b ^ b ^ ,
H ~ ^ = Δ ^ a ^ a ^ + ω a 2 σ ^ z λ ( b ^ + b ^ ) ( a ^ σ ^ + + a ^ σ ^ ) ,
| + , n = cos ( θ 2 ) | g , n + sin ( θ 2 ) | e , n 1 ,
| , n = sin ( θ 2 ) | g , n cos ( θ 2 ) | e , n 1 ,
E ~ g,n = 0 = ω a 2 ,
E ~ n N = [ Δ ^ ( n 1 2 ) ± 1 2 ξ 2 + 4 λ 2 n ( b ^ + b ^ ) ] .
H ^ n N = Δ ^ ( n 1 2 ) I ( n ) + ω m b ^ b ^ + 2 ξ 2 + 4 λ 2 n ( b ^ + b ^ ) σ ^ z ( n ) ,
H ^ M O , e f f ( ± ) = ω m b ^ b ^ ± [ 1 2 ξ 2 + 4 λ 2 n ξ ( n 1 2 ) ] × ( b ^ + b ^ ) + ω c ( n 1 2 ) ,
H ^ M O , e f f = ω m B ^ B ^ ω m β ± 2 .
H ^ M O , e f f = p ^ 2 2 m o + V ^ ± ( q ^ ) ω m β ± 2 ,
| M ± = D ^ ( β ± ) | M ,
M | N + = { exp [ x 2 2 ] ( x ) M N N ! M ! L N M N ( x ) , M N , exp [ x 2 2 ] ( x ) N M M ! N ! L M M N ( x ) , M < N ,
U ^ n N ( t ) = exp [ i ω c t ( n 1 2 ) I ( n ) ] exp [ i Θ ^ ( t ) ] × exp [ η ^ ( t ) b ^ h . c . ] exp [ i ω m t b ^ b ^ ] ,
| ψ ( 0 ) = | g | n | 0 .
| ψ ( t ) = exp [ i ω c t ( n 1 2 ) I ( n ) ] exp [ i Θ ^ ( t ) ] × exp [ η ^ ( t ) b ^ h . c . ] exp [ i ω m t b ^ b ^ ] | g | n | 0 .
| ψ ( t ) = exp [ i ω c t ( n 1 2 ) ] [ cos ( θ 2 ) exp [ i Θ + ( t ) ] | + , n × | η + ( t ) + sin ( θ 2 ) exp [ i Θ ( t ) ] | , n | η ( t ) ] ,
| ψ ( t ) = exp [ i ω c ( n 1 2 ) t ] ( N + 1 | g , n | χ + ( t ) m + N 1 | e , n 1 | χ ( t ) m ) ,
| χ + ( t ) m = N + ( cos 2 ( θ 2 ) exp [ i Θ + ( t ) ] | η + ( t ) + sin 2 ( θ 2 ) exp [ i Θ ( t ) ] | η ( t ) ) ,
| χ ( t ) m = N sin θ 2 ( exp [ i Θ + ( t ) ] | η + ( t ) exp [ i Θ ( t ) ] | η ( t ) ) ,
N + ( t ) = { 1 1 2 sin 2 θ ( [ e i ( Θ + ( t ) Θ ( t ) ) η + ( t ) η ( t ) ] 1 e 1 2 ( | η + ( t ) | 2 + | η ( t ) | 2 ) × R e [ e i ( Θ + ( t ) Θ ( t ) ) η + ( t ) η ( t ) ] ) } 1 / 2 ,
N ( t ) = ( sin θ 2 ) 1 { 2 e 1 2 ( | η + ( t ) | 2 + | η ( t ) | 2 ) × R e [ e i ( Θ + ( t ) Θ ( t ) ) η + ( t ) η ( t ) ] } 1 / 2 ,
N ABC ( ρ ^ ) = ( N A B C N B A C N C A B ) 1 / 3 ,
ρ ^ T a = | N + | 2 | g , n g , n | | χ + ( t ) m χ + ( t ) | m + | N | 2 × | e , n 1 e , n 1 | | χ ( t ) m χ ( t ) | m + N + 1 ( N ) 1 × | e , n g , n 1 | | χ + ( t ) m χ ( t ) | m + N 1 ( N + ) 1 × | g , n 1 e , n | | χ ( t ) m χ + ( t ) | m ,
N A F M ( ρ ^ ) = 2 | N + ( t ) | | N ( t ) | .
W m ( ± ) ( α , α ) = 2 π T r [ D ^ ( α ) ρ ^ m ( ± ) ( t ) D ^ ( α ) ( 1 ) b ^ b ^ ] ,
ρ ^ ˙ ( t ) = i [ H ^ , ρ ^ ] + γ m 2 ( n t h + 1 ) L ( b ^ ) ρ ^ + γ m 2 n t h L ( b ^ ) ρ ^ ,
| ψ ( 0 ) = | g | n | 0 = ( cos ( θ 2 ) | + , n + sin ( θ 2 ) | , n ) | 0 .
U ^ n N ( t ) = exp [ i H ^ n N t / ] = exp [ i ω c ( n 1 2 ) I ( n ) t ] × exp [ i ω m t { b ^ b ^ + β ^ z ( n ) ( b ^ + b ^ ) } ] ,
T ^ U ^ n N ( t ) T ^ = exp [ i ω c ( n 1 2 ) I ( n ) t ] exp [ i ω m t b ^ b ^ ] × exp [ i ω m t ( β ^ z ( n ) ) 2 ] ,
T ^ f ( b ^ , b ^ ) T ^ = f ( T ^ b ^ T ^ , T ^ b ^ T ^ ) ,
T ^ b ^ T ^ = b ^ β ^ z ( n ) ,
T ^ b ^ T ^ = b ^ β ^ z ( n ) .
U ^ n N ( t ) = T ^ T ^ U ^ n N ( t ) T ^ T ^ = exp [ i ω c t ( n 1 2 ) I ( n ) ] × exp [ i ω m t ( β ^ z ( n ) ) 2 ] T ^ T ^ ( β ^ z ( n ) e i ω m t ) × exp [ i ω m t b ^ b ^ ] ,
T ( β ^ z ( n ) e i ω m t ) = exp [ β ^ z ( n ) ( b ^ e i ω m t c . c . ) ] = exp [ i ω m t b ^ b ^ ] T ^ exp [ i ω m t b ^ b ^ ] .
T ^ ( β ^ z ( n ) ) T ( β ^ z ( n ) e i ω m t ) = exp [ β ^ z ( n ) ( b ^ ( 1 e i ω m t ) h . c . ) ] × exp [ i ( β ^ z ( n ) ) 2 sin ( ω m t ) ] .
U ^ n N ( t ) = exp [ i ω c t ( n 1 2 ) I ( n ) ] exp [ i Θ ^ ( t ) ] × exp [ η ^ ( t ) b ^ h . c . ] exp [ i ω m t b ^ b ^ ] ,