Abstract

We propose a scheme to generate the macroscopic entangled Schrödinger cat state of two mechanical resonators in a hybrid optomechanical system by introducing modulated photon-hopping interaction between two cavities. We show that the obtained entangled cat state possesses large average phonon number and the two base vectors of which are nearly orthogonal, thus causing the high degree of entanglement. To justify the robust of the scheme, the dissipation of the system and the noise from the environment are considered. It shows that the high fidelity between the obtained state and the target state can be achieved in the presence of dissipation and thermal noise within our parameter regions, which suggests that our state-generation proposal is feasible.

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

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References

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    [Crossref]
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    [Crossref]
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  41. B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Optomechanical quadrature squeezing in the non-markovian regime,” Opt. Lett. 43, 6053–6056 (2018).
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  42. B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Quantum transistor with a double-cavity optomechanical system,” EPL (Europhysics Lett. 122, 64002 (2018).
    [Crossref]
  43. X.-W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
    [Crossref]
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    [Crossref]
  45. C.-H. Bai, D.-Y. Wang, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Reports 7, 2545 (2017).
    [Crossref]
  46. 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]
  47. C.-G. Liao, H. Xie, X. Shang, Z.-H. Chen, and X.-M. Lin, “Enhancement of steady-state bosonic squeezing and entanglement in a dissipative optomechanical system,” Opt. Express 26, 13783–13799 (2018).
    [Crossref] [PubMed]
  48. J.-Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116, 163602 (2016).
    [Crossref] [PubMed]
  49. 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]
  50. Y.-X. Zeng, T. Gebremariam, M.-S. Ding, and C. Li, “Quantum optical diode based on lyapunov control in a superconducting system,” J. Opt. Soc. Am. B 35, 2334–2341 (2018).
    [Crossref]
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    [Crossref]
  52. D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Entangling a nanomechanical resonator and a superconducting microwave cavity,” Phys. Rev. A 76, 042336 (2007).
    [Crossref]
  53. B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Annals Phys. 385, 757–768 (2017).
    [Crossref]
  54. J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
    [Crossref]
  55. T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the josephson effect,” Phys. Rev. Lett. 112, 203603 (2014).
    [Crossref]

2019 (1)

2018 (16)

Z.-C. Zhang, Y.-P. Wang, Y.-F. Yu, and Z.-M. Zhang, “Quantum squeezing in a modulated optomechanical system,” Opt. Express 26, 11915–11927 (2018).
[Crossref] [PubMed]

C.-G. Liao, H. Xie, X. Shang, Z.-H. Chen, and X.-M. Lin, “Enhancement of steady-state bosonic squeezing and entanglement in a dissipative optomechanical system,” Opt. Express 26, 13783–13799 (2018).
[Crossref] [PubMed]

Y.-X. Zeng, T. Gebremariam, M.-S. Ding, and C. Li, “Quantum optical diode based on lyapunov control in a superconducting system,” J. Opt. Soc. Am. B 35, 2334–2341 (2018).
[Crossref]

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Optomechanical quadrature squeezing in the non-markovian regime,” Opt. Lett. 43, 6053–6056 (2018).
[Crossref] [PubMed]

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]

J. Huang, M. Zhuang, B. Lu, Y. Ke, and C. Lee, “Achieving heisenberg-limited metrology with spin cat states via interaction-based readout,” Phys. Rev. A 98, 012129 (2018).
[Crossref]

T. Hatomura, “Shortcuts to adiabatic cat-state generation in bosonic josephson junctions,” New J. Phys. 20, 015010 (2018).
[Crossref]

J. Cohn, A. Safavi-Naini, R. J. Lewis-Swan, J. G. Bohnet, M. Gärttner, K. A. Gilmore, J. E. Jordan, A. M. Rey, J. J. Bollinger, and J. K. Freericks, “Bang-bang shortcut to adiabaticity in the dicke model as realized in a penning trap experiment,” New J. Phys. 20, 055013 (2018).
[Crossref]

M. Khazali, “Progress towards macroscopic spin and mechanical superposition via rydberg interaction,” Phys. Rev. A 98, 043836 (2018).
[Crossref]

T. Serikawa, J.-i. Yoshikawa, S. Takeda, H. Yonezawa, T. C. Ralph, E. H. Huntington, and A. Furusawa, “Generation of a cat state in an optical sideband,” Phys. Rev. Lett. 121, 143602 (2018).
[Crossref] [PubMed]

Z.-R. Zhong, X.-J. Huang, Z.-B. Yang, L.-T. Shen, and S.-B. Zheng, “Generation and stabilization of entangled coherent states for the vibrational modes of a trapped ion,” Phys. Rev. A 98, 032311 (2018).
[Crossref]

X.-Y. Lü, G.-L. Zhu, L.-L. Zheng, and Y. Wu, “Entanglement and quantum superposition induced by a single photon,” Phys. Rev. A 97, 033807 (2018).
[Crossref]

Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97, 023841 (2018).
[Crossref]

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Quantum transistor with a double-cavity optomechanical system,” EPL (Europhysics Lett. 122, 64002 (2018).
[Crossref]

X.-W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

J. Cheng, X.-T. Liang, W.-Z. Zhang, and X. Duan, “Optomechanical state transfer between two distant membranes in the presence of non-markovian environments,” Chin. Phys. B 27, 120302 (2018).
[Crossref]

2017 (6)

C.-H. Bai, D.-Y. Wang, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Reports 7, 2545 (2017).
[Crossref]

R. Guo, L. Zhou, S.-P. Gu, X.-F. Wang, and Y.-B. Sheng, “Generation of concatenated greenberger–horne–zeilinger-type entangled coherent state based on linear optics,” Quantum Inf. Process. 16, 68 (2017).
[Crossref]

K. Johnson, J. Wong-Campos, B. Neyenhuis, J. Mizrahi, and C. Monroe, “Ultrafast creation of large schrödinger cat states of an atom,” Nat. communications 8, 697 (2017).
[Crossref]

V. Montenegro, R. Coto, V. Eremeev, and M. Orszag, “Macroscopic nonclassical-state preparation via postselection,” Phys. Rev. A 96, 053851 (2017).
[Crossref]

B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Annals Phys. 385, 757–768 (2017).
[Crossref]

W. Asavanant, K. Nakashima, Y. Shiozawa, J.-I. Yoshikawa, and A. Furusawa, “Generation of highly pure schrödinger’s cat states and real-time quadrature measurements via optical filtering,” Opt. Express 25, 32227–32242 (2017).
[Crossref]

2016 (7)

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

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]

W.-W. Zhang, S. K. Goyal, F. Gao, B. C. Sanders, and C. Simon, “Creating cat states in one-dimensional quantum walks using delocalized initial states,” New J. Phys. 18, 093025 (2016).
[Crossref]

J. Joo and E. Ginossar, “Efficient scheme for hybrid teleportation via entangled coherent states in circuit quantum electrodynamics,” Sci. reports 6, 26338 (2016).
[Crossref]

W. Chao-Ping, X.-Y. Hu, Y.-F. Yu, and Z.-M. Zhang, “Phase sensitivity of two nonlinear interferometers with inputting entangled coherent states,” Chin. Phys. B 25, 040601 (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] [PubMed]

T. Liu, Q.-P. Su, S.-J. Xiong, J.-M. Liu, C.-P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit qed,” Sci. reports 6, 32004 (2016).
[Crossref]

2015 (2)

J.-Q. Zhang, W. Xiong, S. Zhang, Y. Li, and M. Feng, “Generating the schrödinger cat state in a nanomechanical resonator coupled to a charge qubit,” Annalen der Physik 527, 180–186 (2015).
[Crossref]

Y.-C. Liu, R.-S. Liu, C.-H. Dong, Y. Li, Q. Gong, and Y.-F. Xiao, “Cooling mechanical resonators to the quantum ground state from room temperature,” Phys. Rev. A 91, 013824 (2015).
[Crossref]

2014 (3)

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

H. Tan, “Deterministic quantum superpositions and fock states of mechanical oscillators via quantum interference in single-photon cavity optomechanics,” Phys. Rev. A 89, 053829 (2014).
[Crossref]

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the josephson effect,” Phys. Rev. Lett. 112, 203603 (2014).
[Crossref]

2013 (4)

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]

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

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

Z.-q. Yin, T. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88, 033614 (2013).
[Crossref]

2010 (1)

J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

2008 (1)

L. Zhou and H. Xiong, “A macroscopical entangled coherent state generator in a v configuration atom system,” J. Phys. B: At. Mol. Opt. Phys. 41, 025501 (2008).
[Crossref]

2007 (2)

I. Katz, A. Retzker, R. Straub, and R. Lifshitz, “Signatures for a classical to quantum transition of a driven nonlinear nanomechanical resonator,” Phys. Rev. Lett. 99, 040404 (2007).
[Crossref] [PubMed]

D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Entangling a nanomechanical resonator and a superconducting microwave cavity,” Phys. Rev. A 76, 042336 (2007).
[Crossref]

2003 (2)

M. Paternostro, M. S. Kim, and B. S. Ham, “Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime,” Phys. Rev. A 67, 023811 (2003).
[Crossref]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[Crossref]

2002 (1)

X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2002).
[Crossref]

2001 (2)

S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 64, 022313 (2001).
[Crossref]

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

2000 (1)

D. Vitali, M. Fortunat, P. Tombesi, and F. De Martini, “Generating entangled schrödinger cat states within a parametric oscillator,” Fortschritte der Physik: Prog. Phys. 48, 437–446 (2000).
[Crossref]

1996 (1)

D. Mogilevtsev and K. S. Ya, “The generation of multicomponent entangled schrödinger cat states via a fully quantized nondegenerate four-wave mixing process,” Opt. Commun. 132, 452–456 (1996).
[Crossref]

1992 (1)

V. Bužek, A. Vidiella-Barranco, and P. L. Knight, “Superpositions of coherent states: Squeezing and dissipation,” Phys. Rev. A 45, 6570–6585 (1992).
[Crossref]

1986 (1)

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref] [PubMed]

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]

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).
[Crossref]

Allman, M. S.

J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Asavanant, W.

Aspelmeyer, M.

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

Bai, C.-H.

C.-H. Bai, D.-Y. Wang, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Reports 7, 2545 (2017).
[Crossref]

Bariani, F.

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

Bohnet, J. G.

J. Cohn, A. Safavi-Naini, R. J. Lewis-Swan, J. G. Bohnet, M. Gärttner, K. A. Gilmore, J. E. Jordan, A. M. Rey, J. J. Bollinger, and J. K. Freericks, “Bang-bang shortcut to adiabaticity in the dicke model as realized in a penning trap experiment,” New J. Phys. 20, 055013 (2018).
[Crossref]

Bollinger, J. J.

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T. Serikawa, J.-i. Yoshikawa, S. Takeda, H. Yonezawa, T. C. Ralph, E. H. Huntington, and A. Furusawa, “Generation of a cat state in an optical sideband,” Phys. Rev. Lett. 121, 143602 (2018).
[Crossref] [PubMed]

Shang, X.

Shen, L.-T.

Z.-R. Zhong, X.-J. Huang, Z.-B. Yang, L.-T. Shen, and S.-B. Zheng, “Generation and stabilization of entangled coherent states for the vibrational modes of a trapped ion,” Phys. Rev. A 98, 032311 (2018).
[Crossref]

Sheng, Y.-B.

R. Guo, L. Zhou, S.-P. Gu, X.-F. Wang, and Y.-B. Sheng, “Generation of concatenated greenberger–horne–zeilinger-type entangled coherent state based on linear optics,” Quantum Inf. Process. 16, 68 (2017).
[Crossref]

Shi, Z.-C.

Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97, 023841 (2018).
[Crossref]

Shiozawa, Y.

Sillanpää, M. A.

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the josephson effect,” Phys. Rev. Lett. 112, 203603 (2014).
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Simmonds, R. W.

J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

Simon, C.

W.-W. Zhang, S. K. Goyal, F. Gao, B. C. Sanders, and C. Simon, “Creating cat states in one-dimensional quantum walks using delocalized initial states,” New J. Phys. 18, 093025 (2016).
[Crossref]

Sirois, A. J.

J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

Song, J.

Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97, 023841 (2018).
[Crossref]

Song, L. N.

X.-W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

Stoler, D.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref] [PubMed]

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I. Katz, A. Retzker, R. Straub, and R. Lifshitz, “Signatures for a classical to quantum transition of a driven nonlinear nanomechanical resonator,” Phys. Rev. Lett. 99, 040404 (2007).
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T. Liu, Q.-P. Su, S.-J. Xiong, J.-M. Liu, C.-P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit qed,” Sci. reports 6, 32004 (2016).
[Crossref]

Su, S.

Su, X.

M. Wang, Z. Qin, M. Zhang, L. Zeng, X. Su, C. Xie, and K. Peng, “Amplifying schrödinger cat state with an optical parametric amplifier,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2018), pp. FTh4G–7.

Takeda, S.

T. Serikawa, J.-i. Yoshikawa, S. Takeda, H. Yonezawa, T. C. Ralph, E. H. Huntington, and A. Furusawa, “Generation of a cat state in an optical sideband,” Phys. Rev. Lett. 121, 143602 (2018).
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H. Tan, “Deterministic quantum superpositions and fock states of mechanical oscillators via quantum interference in single-photon cavity optomechanics,” Phys. Rev. A 89, 053829 (2014).
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H. Tan, F. Bariani, G. Li, and P. Meystre, “Generation of macroscopic quantum superpositions of optomechanical oscillators by dissipation,” Phys. Rev. A 88, 023817 (2013).
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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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Teufel, J. D.

J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
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J.-Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116, 163602 (2016).
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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).
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D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Entangling a nanomechanical resonator and a superconducting microwave cavity,” Phys. Rev. A 76, 042336 (2007).
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D. Vitali, M. Fortunat, P. Tombesi, and F. De Martini, “Generating entangled schrödinger cat states within a parametric oscillator,” Fortschritte der Physik: Prog. Phys. 48, 437–446 (2000).
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Tuorila, J.

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the josephson effect,” Phys. Rev. Lett. 112, 203603 (2014).
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V. Bužek, A. Vidiella-Barranco, and P. L. Knight, “Superpositions of coherent states: Squeezing and dissipation,” Phys. Rev. A 45, 6570–6585 (1992).
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Vitali, D.

D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Entangling a nanomechanical resonator and a superconducting microwave cavity,” Phys. Rev. A 76, 042336 (2007).
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D. Vitali, M. Fortunat, P. Tombesi, and F. De Martini, “Generating entangled schrödinger cat states within a parametric oscillator,” Fortschritte der Physik: Prog. Phys. 48, 437–446 (2000).
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C.-H. Bai, D.-Y. Wang, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Reports 7, 2545 (2017).
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Wang, H.-F.

C.-H. Bai, D.-Y. Wang, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Reports 7, 2545 (2017).
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Wang, M.

M. Wang, Z. Qin, M. Zhang, L. Zeng, X. Su, C. Xie, and K. Peng, “Amplifying schrödinger cat state with an optical parametric amplifier,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2018), pp. FTh4G–7.

Wang, X.

X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2002).
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X. Wang, “Quantum teleportation of entangled coherent states,” Phys. Rev. A 64, 022302 (2001).
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Wang, X.-F.

R. Guo, L. Zhou, S.-P. Gu, X.-F. Wang, and Y.-B. Sheng, “Generation of concatenated greenberger–horne–zeilinger-type entangled coherent state based on linear optics,” Quantum Inf. Process. 16, 68 (2017).
[Crossref]

Wang, X.-Y.

B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Annals Phys. 385, 757–768 (2017).
[Crossref]

Wang, Y.-P.

Wang, Z. H.

X.-W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

Whittaker, J. D.

J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
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Wong-Campos, J.

K. Johnson, J. Wong-Campos, B. Neyenhuis, J. Mizrahi, and C. Monroe, “Ultrafast creation of large schrödinger cat states of an atom,” Nat. communications 8, 697 (2017).
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Woolley, M. J.

D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Entangling a nanomechanical resonator and a superconducting microwave cavity,” Phys. Rev. A 76, 042336 (2007).
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Wu, Y.

X.-Y. Lü, G.-L. Zhu, L.-L. Zheng, and Y. Wu, “Entanglement and quantum superposition induced by a single photon,” Phys. Rev. A 97, 033807 (2018).
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Xia, Y.

Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97, 023841 (2018).
[Crossref]

Xiao, Y.-F.

Y.-C. Liu, R.-S. Liu, C.-H. Dong, Y. Li, Q. Gong, and Y.-F. Xiao, “Cooling mechanical resonators to the quantum ground state from room temperature,” Phys. Rev. A 91, 013824 (2015).
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Xie, C.

M. Wang, Z. Qin, M. Zhang, L. Zeng, X. Su, C. Xie, and K. Peng, “Amplifying schrödinger cat state with an optical parametric amplifier,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2018), pp. FTh4G–7.

Xie, H.

Xiong, B.

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Optomechanical quadrature squeezing in the non-markovian regime,” Opt. Lett. 43, 6053–6056 (2018).
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B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Quantum transistor with a double-cavity optomechanical system,” EPL (Europhysics Lett. 122, 64002 (2018).
[Crossref]

B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Annals Phys. 385, 757–768 (2017).
[Crossref]

Xiong, H.

L. Zhou and H. Xiong, “A macroscopical entangled coherent state generator in a v configuration atom system,” J. Phys. B: At. Mol. Opt. Phys. 41, 025501 (2008).
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Xiong, S.-J.

T. Liu, Q.-P. Su, S.-J. Xiong, J.-M. Liu, C.-P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit qed,” Sci. reports 6, 32004 (2016).
[Crossref]

Xiong, W.

J.-Q. Zhang, W. Xiong, S. Zhang, Y. Li, and M. Feng, “Generating the schrödinger cat state in a nanomechanical resonator coupled to a charge qubit,” Annalen der Physik 527, 180–186 (2015).
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Xu, X.-W.

X.-W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

Ya, K. S.

D. Mogilevtsev and K. S. Ya, “The generation of multicomponent entangled schrödinger cat states via a fully quantized nondegenerate four-wave mixing process,” Opt. Commun. 132, 452–456 (1996).
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Yan, L.

Yang, C.-P.

T. Liu, Q.-P. Su, S.-J. Xiong, J.-M. Liu, C.-P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit qed,” Sci. reports 6, 32004 (2016).
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Yang, W.

Yang, Z.-B.

Z.-R. Zhong, X.-J. Huang, Z.-B. Yang, L.-T. Shen, and S.-B. Zheng, “Generation and stabilization of entangled coherent states for the vibrational modes of a trapped ion,” Phys. Rev. A 98, 032311 (2018).
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Yin, Z.-q.

Z.-q. Yin, T. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88, 033614 (2013).
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Yonezawa, H.

T. Serikawa, J.-i. Yoshikawa, S. Takeda, H. Yonezawa, T. C. Ralph, E. H. Huntington, and A. Furusawa, “Generation of a cat state in an optical sideband,” Phys. Rev. Lett. 121, 143602 (2018).
[Crossref] [PubMed]

Yoshikawa, J.-i.

T. Serikawa, J.-i. Yoshikawa, S. Takeda, H. Yonezawa, T. C. Ralph, E. H. Huntington, and A. Furusawa, “Generation of a cat state in an optical sideband,” Phys. Rev. Lett. 121, 143602 (2018).
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W. Asavanant, K. Nakashima, Y. Shiozawa, J.-I. Yoshikawa, and A. Furusawa, “Generation of highly pure schrödinger’s cat states and real-time quadrature measurements via optical filtering,” Opt. Express 25, 32227–32242 (2017).
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Yu, Y.-F.

Z.-C. Zhang, Y.-P. Wang, Y.-F. Yu, and Z.-M. Zhang, “Quantum squeezing in a modulated optomechanical system,” Opt. Express 26, 11915–11927 (2018).
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W. Chao-Ping, X.-Y. Hu, Y.-F. Yu, and Z.-M. Zhang, “Phase sensitivity of two nonlinear interferometers with inputting entangled coherent states,” Chin. Phys. B 25, 040601 (2016).
[Crossref]

Yurke, B.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref] [PubMed]

Zeng, L.

M. Wang, Z. Qin, M. Zhang, L. Zeng, X. Su, C. Xie, and K. Peng, “Amplifying schrödinger cat state with an optical parametric amplifier,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2018), pp. FTh4G–7.

Zeng, Y.-X.

Zhang, J.-Q.

J.-Q. Zhang, W. Xiong, S. Zhang, Y. Li, and M. Feng, “Generating the schrödinger cat state in a nanomechanical resonator coupled to a charge qubit,” Annalen der Physik 527, 180–186 (2015).
[Crossref]

Zhang, M.

M. Wang, Z. Qin, M. Zhang, L. Zeng, X. Su, C. Xie, and K. Peng, “Amplifying schrödinger cat state with an optical parametric amplifier,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2018), pp. FTh4G–7.

Zhang, S.

C.-H. Bai, D.-Y. Wang, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Reports 7, 2545 (2017).
[Crossref]

J.-Q. Zhang, W. Xiong, S. Zhang, Y. Li, and M. Feng, “Generating the schrödinger cat state in a nanomechanical resonator coupled to a charge qubit,” Annalen der Physik 527, 180–186 (2015).
[Crossref]

Zhang, W.-W.

W.-W. Zhang, S. K. Goyal, F. Gao, B. C. Sanders, and C. Simon, “Creating cat states in one-dimensional quantum walks using delocalized initial states,” New J. Phys. 18, 093025 (2016).
[Crossref]

Zhang, W.-Z.

J. Cheng, X.-T. Liang, W.-Z. Zhang, and X. Duan, “Optomechanical state transfer between two distant membranes in the presence of non-markovian environments,” Chin. Phys. B 27, 120302 (2018).
[Crossref]

Zhang, X.

Z.-q. Yin, T. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88, 033614 (2013).
[Crossref]

Zhang, Z.-C.

Zhang, Z.-M.

Z.-C. Zhang, Y.-P. Wang, Y.-F. Yu, and Z.-M. Zhang, “Quantum squeezing in a modulated optomechanical system,” Opt. Express 26, 11915–11927 (2018).
[Crossref] [PubMed]

W. Chao-Ping, X.-Y. Hu, Y.-F. Yu, and Z.-M. Zhang, “Phase sensitivity of two nonlinear interferometers with inputting entangled coherent states,” Chin. Phys. B 25, 040601 (2016).
[Crossref]

Zheng, L.-L.

X.-Y. Lü, G.-L. Zhu, L.-L. Zheng, and Y. Wu, “Entanglement and quantum superposition induced by a single photon,” Phys. Rev. A 97, 033807 (2018).
[Crossref]

Zheng, Q.

X.-W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

Zheng, S.-B.

Z.-R. Zhong, X.-J. Huang, Z.-B. Yang, L.-T. Shen, and S.-B. Zheng, “Generation and stabilization of entangled coherent states for the vibrational modes of a trapped ion,” Phys. Rev. A 98, 032311 (2018).
[Crossref]

Zhong, Z.-R.

Z.-R. Zhong, X.-J. Huang, Z.-B. Yang, L.-T. Shen, and S.-B. Zheng, “Generation and stabilization of entangled coherent states for the vibrational modes of a trapped ion,” Phys. Rev. A 98, 032311 (2018).
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Zhou, L.

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Quantum transistor with a double-cavity optomechanical system,” EPL (Europhysics Lett. 122, 64002 (2018).
[Crossref]

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Optomechanical quadrature squeezing in the non-markovian regime,” Opt. Lett. 43, 6053–6056 (2018).
[Crossref] [PubMed]

B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Annals Phys. 385, 757–768 (2017).
[Crossref]

R. Guo, L. Zhou, S.-P. Gu, X.-F. Wang, and Y.-B. Sheng, “Generation of concatenated greenberger–horne–zeilinger-type entangled coherent state based on linear optics,” Quantum Inf. Process. 16, 68 (2017).
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L. Zhou and H. Xiong, “A macroscopical entangled coherent state generator in a v configuration atom system,” J. Phys. B: At. Mol. Opt. Phys. 41, 025501 (2008).
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C.-H. Bai, D.-Y. Wang, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Reports 7, 2545 (2017).
[Crossref]

Zhu, G.-L.

X.-Y. Lü, G.-L. Zhu, L.-L. Zheng, and Y. Wu, “Entanglement and quantum superposition induced by a single photon,” Phys. Rev. A 97, 033807 (2018).
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Zhuang, M.

J. Huang, M. Zhuang, B. Lu, Y. Ke, and C. Lee, “Achieving heisenberg-limited metrology with spin cat states via interaction-based readout,” Phys. Rev. A 98, 012129 (2018).
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Annalen der Physik (1)

J.-Q. Zhang, W. Xiong, S. Zhang, Y. Li, and M. Feng, “Generating the schrödinger cat state in a nanomechanical resonator coupled to a charge qubit,” Annalen der Physik 527, 180–186 (2015).
[Crossref]

Annals Phys. (1)

B. Xiong, X. Li, X.-Y. Wang, and L. Zhou, “Improve microwave quantum illumination via optical parametric amplifier,” Annals Phys. 385, 757–768 (2017).
[Crossref]

Chin. Phys. B (2)

J. Cheng, X.-T. Liang, W.-Z. Zhang, and X. Duan, “Optomechanical state transfer between two distant membranes in the presence of non-markovian environments,” Chin. Phys. B 27, 120302 (2018).
[Crossref]

W. Chao-Ping, X.-Y. Hu, Y.-F. Yu, and Z.-M. Zhang, “Phase sensitivity of two nonlinear interferometers with inputting entangled coherent states,” Chin. Phys. B 25, 040601 (2016).
[Crossref]

EPL (Europhysics Lett. (1)

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Quantum transistor with a double-cavity optomechanical system,” EPL (Europhysics Lett. 122, 64002 (2018).
[Crossref]

Fortschritte der Physik: Prog. Phys. (1)

D. Vitali, M. Fortunat, P. Tombesi, and F. De Martini, “Generating entangled schrödinger cat states within a parametric oscillator,” Fortschritte der Physik: Prog. Phys. 48, 437–446 (2000).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. A: Math. Gen. (1)

X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2002).
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J. Phys. B: At. Mol. Opt. Phys. (2)

Y. Chen, “Macroscopic quantum mechanics: theory and experimental concepts of optomechanics,” J. Phys. B: At. Mol. Opt. Phys. 46, 104001 (2013).
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L. Zhou and H. Xiong, “A macroscopical entangled coherent state generator in a v configuration atom system,” J. Phys. B: At. Mol. Opt. Phys. 41, 025501 (2008).
[Crossref]

Nat. communications (1)

K. Johnson, J. Wong-Campos, B. Neyenhuis, J. Mizrahi, and C. Monroe, “Ultrafast creation of large schrödinger cat states of an atom,” Nat. communications 8, 697 (2017).
[Crossref]

Nature (1)

J. D. Teufel, L. Dale, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
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New J. Phys. (4)

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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T. Hatomura, “Shortcuts to adiabatic cat-state generation in bosonic josephson junctions,” New J. Phys. 20, 015010 (2018).
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J. Cohn, A. Safavi-Naini, R. J. Lewis-Swan, J. G. Bohnet, M. Gärttner, K. A. Gilmore, J. E. Jordan, A. M. Rey, J. J. Bollinger, and J. K. Freericks, “Bang-bang shortcut to adiabaticity in the dicke model as realized in a penning trap experiment,” New J. Phys. 20, 055013 (2018).
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W.-W. Zhang, S. K. Goyal, F. Gao, B. C. Sanders, and C. Simon, “Creating cat states in one-dimensional quantum walks using delocalized initial states,” New J. Phys. 18, 093025 (2016).
[Crossref]

Opt. Commun. (1)

D. Mogilevtsev and K. S. Ya, “The generation of multicomponent entangled schrödinger cat states via a fully quantized nondegenerate four-wave mixing process,” Opt. Commun. 132, 452–456 (1996).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (19)

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]

D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, “Entangling a nanomechanical resonator and a superconducting microwave cavity,” Phys. Rev. A 76, 042336 (2007).
[Crossref]

Y.-C. Liu, R.-S. Liu, C.-H. Dong, Y. Li, Q. Gong, and Y.-F. Xiao, “Cooling mechanical resonators to the quantum ground state from room temperature,” Phys. Rev. A 91, 013824 (2015).
[Crossref]

Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97, 023841 (2018).
[Crossref]

H. Tan, “Deterministic quantum superpositions and fock states of mechanical oscillators via quantum interference in single-photon cavity optomechanics,” Phys. Rev. A 89, 053829 (2014).
[Crossref]

X.-W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

X.-Y. Lü, G.-L. Zhu, L.-L. Zheng, and Y. Wu, “Entanglement and quantum superposition induced by a single photon,” Phys. Rev. A 97, 033807 (2018).
[Crossref]

V. Bužek, A. Vidiella-Barranco, and P. L. Knight, “Superpositions of coherent states: Squeezing and dissipation,” Phys. Rev. A 45, 6570–6585 (1992).
[Crossref]

V. Montenegro, R. Coto, V. Eremeev, and M. Orszag, “Macroscopic nonclassical-state preparation via postselection,” Phys. Rev. A 96, 053851 (2017).
[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]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[Crossref]

J. Huang, M. Zhuang, B. Lu, Y. Ke, and C. Lee, “Achieving heisenberg-limited metrology with spin cat states via interaction-based readout,” Phys. Rev. A 98, 012129 (2018).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 (a) and (b) plot |Jm(2ξ)| as a function of 2ξ and m , respectively. (c) and (d) are the fidelity as a function of the time t and frequency ωm (in units of g0) at δ = g, respectively. Fidelity F(t0) corresponding to ωm/g0 ranging from 26 to 60 is enlarged in the subgraph of (d), where t0 = π/δ.
Fig. 2
Fig. 2 Plot of time dependence of the average excitation phonon number b in (a) and (b), and entanglement in (c) and (d). The other parameters are n0 = 1, ξ = 1.5271 and δ = g.
Fig. 3
Fig. 3 Plot of the time dependence of fidelity in (a), (b), (c) and entanglement in (d) at ωm/g0 = 50. The cavity decay κ and dissipation γ labeled in figures are in units of g0. The other parameters are n0 = 1, ξ = 1.5271 and δ = g.
Fig. 4
Fig. 4 (a) the time dependence of average excitation number Nb(t) with different cavity loss κ. (b) average excitation number Nb(t) as a function a κ at the target time t = tf. Here γ = 10−3 and th = 1 for (a) and (b), and the other parameters see Fig. 2.

Equations (27)

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H = j = 1 , 2 [ ω c a j a j + ω m b j b j + g 0 a j a j ( b j + b j ) ] ξ ω 0 cos ( ω 0 t ) ( a 1 a 2 + a 2 a 1 ) ,
a ± = 1 2 ( a 1 ± a 2 ) , b ± = 1 2 ( b 1 ± b 2 ) .
H = k = + , [ ω k ( t ) a k a k + ω m b k b k ] + g 0 2 ( a + a + + a a ) × ( b + + b + ) + g 0 2 ( a a + + a + a ) ( b + b ) ,
T ( t ) = exp { i k = ± 0 t d τ [ ω k ( τ ) a k a k + ω m b k b k ] } = exp { i k = ± [ ω ˜ k ( t ) a k a k + ω m t b k b k ] } ,
H ˜ = g 0 2 ( a + a + + a a ) ( b + e i ω m t + b + e i ω m t ) + g 0 2 ( a + a e 2 i ξ sin ( ω 0 t ) + a a + e 2 i ξ sin ( ω 0 t ) ) × ( b e i ω m t + b e i ω m t ) .
H ˜ = g 0 2 ( a + a + + a a ) ( b + e i ω m t + b + e i ω m t ) + g 0 2 J 0 ( 2 ξ ) ( a + a + a a + ) ( b e i ω m t + b e i ω m t ) + g 0 2 n = 1 J 2 n 1 ( 2 ξ ) ( a + a a a + ) [ b e i [ ω m + ( 2 n 1 ) ω 0 t ] b e i [ ω m ( 2 n 1 ) ω 0 ] t h . c . ] + g 0 2 n = 1 J 2 n ( 2 ξ ) ( a + a + a a + ) [ b e i ( ω m + 2 n ω 0 ) t + b e i ( ω m 2 n ω 0 ) t + h . c . ] .
H ˜ = g 0 J 2 n 0 ( 2 ξ ) 2 ( a + a + a a + ) ( b e i δ t + b e i δ t ) .
H ˜ ( t ) = g ( a 1 a 1 a 2 a 2 ) ( b 1 e i δ t b 2 e i δ t + h . c . ) ,
| ψ ˜ ( t ) = U ( t ) | ψ ˜ ( 0 ) = 𝒯 exp { i 0 t H ˜ ( τ ) d τ } | ψ ˜ ( 0 ) ,
U ( t ) = 𝒯 exp { 0 t [ i H ˜ ( τ ) ] d τ } = exp { A 1 + A 2 } ,
A 1 = g δ ( a 1 a 1 a 2 a 2 ) [ ( e i δ t 1 ) ( b 1 b 2 ) h . c . ] , A 2 = 2 i g 2 δ 2 ( a 1 a 1 a 2 a 2 ) 2 ( δ t sin ( δ t ) ) .
| ψ ˜ ( t ) = 1 2 e i χ ˜ ( t ) ( | 1 1 | 0 2 | β ˜ 1 | β ˜ 2 + | 0 1 | 1 2 | β ˜ 1 | β ˜ 2 ) ,
χ ˜ ( t ) = 2 g 2 δ 2 ( δ t sin ( δ t ) ) , β ˜ = g δ ( 1 e i δ t ) .
| ψ ( t ) = 1 2 e i χ ( t ) ( | 1 1 | 0 2 | φ L + | 0 1 | 1 2 | φ R ) ,
| φ L = cos ϑ | β 1 | β 2 + i sin ϑ | β 1 | β 2 , | φ R = cos ϑ | β 1 | β 2 + i sin ϑ | β 1 | β 2 ,
N ¯ b = 2 g 2 δ 2 [ 1 cos ( δ t ) ] .
| φ L , s = 1 2 ( | β 1 | β 2 + i | β 1 | β 2 ) .
N = log 2 ρ L T 1 1 = log 2 Tr [ ( ρ L T 1 ) ρ L T 1 ] ,
E n = log 2 [ ( λ + + λ ) 2 ] .
λ ± = 1 2 ± 1 2 1 ( 1 e 4 | β | 2 ) 2 | sin ( 2 ϑ ) | 2 ,
ρ ˙ = i [ H , ρ ] + j κ j [ a j ] ρ + γ j ( n ¯ j + 1 ) [ b j ] ρ + γ j n ¯ j [ b j ] ρ ,
ρ L = 1 | 2 1 0 | ρ | 1 1 | 0 2 tr [ 1 | 2 1 0 | ρ | 1 1 | 0 2 ] or ρ R = 0 | 2 1 1 | ρ | 0 1 | 1 2 tr [ 0 | 2 1 1 | ρ | 0 1 | 1 2 ]
e i z sin ν = m = J m ( z ) e i m ν ,
e 2 i ξ sin ω 0 t = J 0 ( 2 ξ ) + n = 1 [ J 2 n ( 2 ξ ) ( e i 2 n ω 0 t + e i 2 n ω 0 t ) + J 2 n 1 ( 2 ξ ) ( e i ( 2 n 1 ) ω 0 t e i ( 2 n 1 ) ω 0 t ) ] , e 2 i ξ sin ω 0 t = J 0 ( 2 ξ ) + n = 1 [ J 2 n ( 2 ξ ) ( e i 2 n ω 0 t + e i 2 n ω 0 t ) J 2 n 1 ( 2 ξ ) ( e i ( 2 n 1 ) ω 0 t e i ( 2 n 1 ) ω 0 t ) ] .
U ( t ) = 𝒯 exp { 0 t [ i H ˜ ( τ ) ] d τ } = exp [ A ( t ) ] ,
A 1 ( t ) = 0 t i H ˜ ( τ 1 ) d τ 1 A 2 ( t ) = 1 2 0 t d τ 1 0 τ 1 d τ 2 [ i H ˜ ( τ 1 ) , i H ˜ ( τ 2 ) ] A 3 ( t ) = 1 6 0 t d τ 1 0 τ 1 d τ 2 0 τ 2 d τ 3 ( [ i H ˜ ( τ 1 ) , [ i H ˜ ( τ 2 ) , i H ˜ ( τ 3 ) ] ] + [ i H ˜ ( τ 3 ) , [ i H ˜ ( τ 2 ) , i H ˜ ( τ 1 ) ] ] ) , A k > 3 ( t ) = .
[ i H ˜ ( τ ) , i H ˜ ( τ ) ] = 4 i g 2 ( a 1 a 1 a 2 a 2 ) 2 sin [ δ ( τ τ ] ,

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