Abstract

Previous works for achieving mechanical squeezing focused mainly on the sole squeezing manipulation method. Here we study how to construct strong steady-state mechanical squeezing via the joint effect between Duffing nonlinearity and parametric pump driving. We find that the 3 dB limit of strong mechanical squeezing can be easily overcome from the joint effect of two different below 3 dB squeezing components induced by Duffing nonlinearity and parametric pump driving, respectively, without the need of any extra technologies, such as quantum measurement or quantum feedback. We first demonstrate that, in the ideal mechanical bath, the joint squeezing effect just is the superposition of the two respective independent squeezing components. The mechanical squeezing constructed by the joint effect is fairly robust against the mechanical thermal noise. Moreover, different from previous mechanical squeezing detection schemes, which need to introduce an additional ancillary cavity mode, the joint mechanical squeezing effect in the present scheme can be directly measured by homodyning the output field of the cavity with an appropriate phase. The joint idea opens up a new approach to construct strong mechanical squeezing and can be generalized to realize other strong macroscopic quantum effects.

© 2019 Chinese Laser Press

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  5. M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
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  7. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
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  20. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
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  21. J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
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  23. A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
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  24. J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
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  28. A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).
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  32. M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89, 023849 (2014).
    [Crossref]
  33. S. Huang and G. S. Agarwal, “Reactive coupling can beat the motional quantum limit of nanowaveguides coupled to a microdisk resonator,” Phys. Rev. A 82, 033811 (2010).
    [Crossref]
  34. W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
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    [Crossref]
  43. A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
    [Crossref]
  44. D. Y. Wang, C. H. Bai, S. Liu, S. Zhang, and H. F. Wang, “Optomechanical cooling beyond the quantum backaction limit with frequency modulation,” Phys. Rev. A 98, 023816 (2018).
    [Crossref]
  45. J. F. Huang, J. Q. Liao, and L. M. Kuang, “Ultrastrong Jaynes-Cummings model,” arXiv:1902.05779 (2019).
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    [Crossref]
  47. E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
    [Crossref]
  48. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
    [Crossref]
  49. R. X. Chen, L. T. Shen, Z. B. Yang, and H. Z. Wu, “Transition of entanglement dynamics in an oscillator system with weak time-dependent coupling,” Phys. Rev. A 91, 012312 (2015).
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    [Crossref]

2019 (2)

C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Modulation-based atom-mirror entanglement and mechanical squeezing in an unresolved-sideband optomechanical system,” Ann. Phys. 531, 1800271 (2019).
[Crossref]

R. Zhang, Y. Fang, Y. Y. Wang, S. Chesi, and Y. D. Wang, “Strong mechanical squeezing in an unresolved-sideband optomechanical system,” Phys. Rev. A 99, 043805 (2019).
[Crossref]

2018 (4)

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]

A. Dalafi, M. H. Naderi, and A. Motazedifard, “Effects of quadratic coupling and squeezed vacuum injection in an optomechanical cavity assisted with a Bose-Einstein condensate,” Phys. Rev. A 97, 043619 (2018).
[Crossref]

D. Y. Wang, C. H. Bai, S. Liu, S. Zhang, and H. F. Wang, “Optomechanical cooling beyond the quantum backaction limit with frequency modulation,” Phys. Rev. A 98, 023816 (2018).
[Crossref]

S. Chakraborty and A. K. Sarma, “Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics,” Phys. Rev. A 97, 022336 (2018).
[Crossref]

2017 (3)

K. W. Xiao, N. Zhao, and Z. Q. Yin, “Bistability and squeezing of the librational mode of an optically trapped nanoparticle,” Phys. Rev. A 96, 013837 (2017).
[Crossref]

H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (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]

2015 (2)

R. X. Chen, L. T. Shen, Z. B. Yang, and H. Z. Wu, “Transition of entanglement dynamics in an oscillator system with weak time-dependent coupling,” Phys. Rev. A 91, 012312 (2015).
[Crossref]

X. Y. Lü, J. Q. Liao, L. Tian, and F. Nori, “Steady-state mechanical squeezing in an optomechanical system via Duffing nonlinearity,” Phys. Rev. A 91, 013834 (2015).
[Crossref]

2014 (2)

M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89, 023849 (2014).
[Crossref]

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

2013 (4)

W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
[Crossref]

W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 20423–20440 (2013).
[Crossref]

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).
[Crossref]

A. Szorkovszky, G. A. Brawley, A. C. Doherty, and W. P. Bowen, “Strong thermomechanical squeezing via weak measurement,” Phys. Rev. Lett. 110, 184301 (2013).
[Crossref]

2012 (4)

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New J. Phys. 14, 125005 (2012).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

2011 (4)

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref]

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[Crossref]

A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, “Mechanical squeezing via parametric amplification and weak measurement,” Phys. Rev. Lett. 107, 213603 (2011).
[Crossref]

2010 (4)

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

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

S. Huang and G. S. Agarwal, “Reactive coupling can beat the motional quantum limit of nanowaveguides coupled to a microdisk resonator,” Phys. Rev. A 82, 033811 (2010).
[Crossref]

2009 (4)

K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[Crossref]

A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
[Crossref]

K. Jacobs and A. J. Landahl, “Engineering giant nonlinearities in quantum nanosystems,” Phys. Rev. Lett. 103, 067201 (2009).
[Crossref]

2008 (4)

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref]

2006 (1)

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[Crossref]

2005 (2)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

R. Ruskov, K. Schwab, and A. N. Korotkov, “Squeezing of a nanomechanical resonator by quantum nondemolition measurement and feedback,” Phys. Rev. B 71, 235407 (2005).
[Crossref]

2004 (1)

M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
[Crossref]

1992 (1)

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

1991 (1)

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36–44 (1991).
[Crossref]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

1986 (1)

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[Crossref]

1985 (1)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

1983 (1)

D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
[Crossref]

1980 (1)

C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980).
[Crossref]

1979 (1)

J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detectors,” Phys. Rev. D 19, 1669–1679 (1979).
[Crossref]

Abramovici, A.

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

Agarwal, G. S.

M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89, 023849 (2014).
[Crossref]

S. Huang and G. S. Agarwal, “Reactive coupling can beat the motional quantum limit of nanowaveguides coupled to a microdisk resonator,” Phys. Rev. A 82, 033811 (2010).
[Crossref]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

Althouse, W. E.

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

Asjad, M.

M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89, 023849 (2014).
[Crossref]

Aspelmeyer, M.

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

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[Crossref]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[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).
[Crossref]

Bai, C. H.

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J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
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A. Dalafi, M. H. Naderi, and A. Motazedifard, “Effects of quadratic coupling and squeezed vacuum injection in an optomechanical cavity assisted with a Bose-Einstein condensate,” Phys. Rev. A 97, 043619 (2018).
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Ann. Phys. (1)

C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Modulation-based atom-mirror entanglement and mechanical squeezing in an unresolved-sideband optomechanical system,” Ann. Phys. 531, 1800271 (2019).
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Nat. Phys. (1)

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
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Nature (8)

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
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Opt. Express (1)

Opt. Lett. (1)

Photon. Res. (1)

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Figures (8)

Fig. 1.
Fig. 1. Schematic diagram of the considered optomechanical system. An OPA is placed inside the cavity driven by an external laser field and is pumped by a parametric driving field. Here the movable mirror is coupled to the cavity field via the radiation-pressure interaction and is treated as a quantum-mechanical oscillator with a Duffing nonlinearity.

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Fig. 2.
Fig. 2. Sketch of the physical processes of the joint effect between Duffing nonlinearity and parametric pump driving in construction of strong mechanical squeezing.

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Fig. 3.
Fig. 3. Dependence of (a) the cavity mode phase mean square fluctuation ⟨δY2⟩ and (b) the mechanical mode position mean square fluctuation ⟨δQ2⟩ on the parametric gain G for the parametric phase θ∈[0,12π]. The horizontal dashed line represents the variance of the vacuum state. The frequency of the mechanical mode ωm/(2π)=2.5×106  Hz. Other parameters are ωc=2.5×108ωm, γm=10−6ωm, κ=0.1ωm, g0=10−4ωm, P=0.1  mW, nmth=ncth=0, and εL=2Pκ/(ℏωc).

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Fig. 4.
Fig. 4. Dependence of the mechanical mode position mean square fluctuation ⟨δQ2⟩ on the parametric gain G in the cases of η=0 and η=10−5ωm. The parameter sets (G,η) corresponding to the points A, B, C, and D are (0,0), (0.4κ,0), (0,10−5ωm), and (0.4κ,10−5ωm), respectively. Here we have set θ=0, and other parameters are the same as in Fig. 3. The shadowed blue bottom region corresponds to squeezing below the 3 dB limit.

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Fig. 5.
Fig. 5. Wigner function in the phase space for the mechanical mode. (a), (b), (c), and (d) correspond to the points A, B, C, and D in Fig. 4, respectively. The parameters are the same as in Fig. 4.

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Fig. 6.
Fig. 6. Mechanical mode position mean square fluctuation ⟨δQ2⟩ obtained by the numerical solution in Eq. (25) and the analytical solution in Eq. (44), respectively, in the cases of η=0 and η=10−5ωm. Other parameters are the same as in Fig. 4. The shadowed blue bottom region corresponds to squeezing below the 3 dB limit.

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Fig. 7.
Fig. 7. Dependence of the mechanical mode position mean square fluctuation ⟨δQ2⟩ on the thermal phonon number nmth. Here we have set κ=0.2ωm, η=10−4ωm, G=0.49κ, θ=0, and P=10  mW. Other parameters are the same as in Fig. 3. The shadowed blue bottom region corresponds to squeezing below the 3 dB limit.

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Fig. 8.
Fig. 8. Contour plot of the detection spectrum SZout(ω) of the quadrature fluctuation of the output field versus the frequency ω and the measurement phase angle ϕ when G=0.4κ and η=10−5ωm. Other parameters are the same as in Fig. 4.

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Tables (1)

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Table 1. Applying Either (Both) of the Parametric Pump Driving and the Duffing Nonlinearity, the Sole (Joint) Squeezing Result (in Units of Decibels) of These Two Different Squeezing Methods in Different Parameter Sets of (P,G,η)a

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Equations (49)

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H=δcc†c+ωmb†b+η2(b+b†)4−g0c†c(b+b†)+εL(c†+c)+iG(eiθc†2e−2iω˜m′t−e−iθc2e2iω˜m′t).
[−i(δc−2g0β)−κ]α−iεL=0,16ηβ3+(12η+ωm)β−g0|α|2=0,
b˙=−iω˜mb−2iΛb†+ig(c+c†)−γm2b+γmbin(t),c˙=−iΔcc+ig(b+b†)+2Geiθc†e−2iω˜m′t−κc+2κcin(t),
Δc=δc−2g0β,ω˜m=ωm+2Λ,Λ=3η(4β2+1),g=g0|α|.
⟨bin†(t)bin(t′)⟩=nmthδ(t−t′),⟨bin(t)bin†(t′)⟩=(nmth+1)δ(t−t′),
⟨cin†(t)cin(t′)⟩=ncthδ(t−t′),⟨cin(t)cin†(t′)⟩=(ncth+1)δ(t−t′),
Heff=ω˜mb†b+Δcc†c+Λ(b2+b†2)−g(b+b†)(c+c†)+iG(eiθc†2e−2iω˜m′t−e−iθc2e2iω˜m′t).
r=14 ln(1+4Λωm)
Heff′=S†(r)HeffS(r)=ω˜m′b†b+Δcc†c−g′(b+b†)(c+c†)+iG(eiθc†2e−2iω˜m′t−e−iθc2e2iω˜m′t),
ω˜m′=ωm2+4ωmΛ,g′=g(1+4Λωm)−14.
Heff′′=−g′[e−i(Δc+ω˜m′)tbc+e−i(Δc−ω˜m′)tb†c+ei(Δc−ω˜m′)tc†b+ei(Δc+ω˜m′)tc†b†]+iG[eiθc†2e2i(Δc−ω˜m′)t−e−iθc2e−2i(Δc−ω˜m′)t].
Heff′′=−g′(b†c+c†b)+iG(eiθc†2−e−iθc2).
b˙=ig′c−γm2b+γmbin(t),c˙=ig′b+2Geiθc†−κc+2κcin(t).
δQ=(b+b†)/2,δP=(b−b†)/2i,Qin=(bin+bin†)/2,Pin=(bin−bin†)/2i,
δX=(c+c†)/2,δY=(c−c†)/2i,Xin=(cin+cin†)/2,Yin=(cin−cin†)/2i,
f˙(t)=Mf(t)+n(t),
f(t)=[δQ,δP,δX,δY]T,n(t)=[γmQin,γmPin,2κXin,2κYin]T,
M=[−γm200−g′0−γm2g′00−g′2G cos θ−κ2G sin θg′02G sin θ−(2G cos θ+κ)].
2κ(κ2−4G2)+14γm3+(2κ+γm)(g′2+2κγm)>0,γm2(κ2−4G2)+4g′2(g′2+κγm)>0,2κγm(κ2−4G2)2+[(2κ+γm)2g′2+(4κ+γm)κγm2]×(κ2−4G2)+κγm(2κ+γm)[κγm2+(2κ+32γm)g′2]+γm34[κγm22+(2κ+γm)g′2]>0.
δQ(ω)=A1(ω)Qin(ω)+B1(ω)Pin(ω)+E1(ω)Xin(ω)+F1(ω)Yin(ω),δP(ω)=A2(ω)Qin(ω)+B2(ω)Pin(ω)+E2(ω)Xin(ω)+F2(ω)Yin(ω),
A1(ω)=γmd(ω){[u(ω)2−4G2]ν(ω)+g′2u(ω)+2Gg′2 cos θ},B1(ω)=γmd(ω)2Gg′2 sin θ,E1(ω)=−2κd(ω)2Gg′ sin θν(ω),F1(ω)=2κd(ω)g′{[2G cos θ−u(ω)]ν(ω)−g′2},A2(ω)=γmd(ω)2Gg′2 sin θ,B2(ω)=γmd(ω){[u(ω)ν(ω)+g′2]u(ω)−4G2ν(ω)−2Gg′2 cos θ},E2(ω)=2κd(ω)g′{[2G cos θ+u(ω)]ν(ω)+g′2},F2(ω)=2κd(ω)2Gg′ sin θν(ω),
2πSZ(ω)δ(ω+Ω)=12[⟨δZ(ω)δZ(Ω)⟩+⟨δZ(Ω)δZ(ω)],Z=Q,P.
⟨Qin(ω)Qin(Ω)⟩=⟨Pin(ω)Pin(Ω)⟩=(nmth+12)2πδ(ω+Ω),⟨Qin(ω)Pin(Ω)⟩=−⟨Pin(ω)Qin(Ω)⟩=iπδ(ω+Ω),⟨Xin(ω)Xin(Ω)⟩=⟨Yin(ω)Yin(Ω)⟩=(ncth+12)2πδ(ω+Ω),⟨Xin(ω)Yin(Ω)⟩=−⟨Yin(ω)Xin(Ω)⟩=iπδ(ω+Ω),
SQ(ω)=[A1(ω)A1(−ω)+B1(ω)B1(−ω)](nmth+12)+[E1(ω)E1(−ω)+F1(ω)F1(−ω)](ncth+12),SP(ω)=[A2(ω)A2(−ω)+B2(ω)B2(−ω)](nmth+12)+[E2(ω)E2(−ω)+F2(ω)F2(−ω)](ncth+12).
⟨δQ2⟩=e−2r2π∫−∞∞SQ(ω)dω,⟨δP2⟩=e2r2π∫−∞∞SP(ω)dω.
δX(ω)=E3(ω)Xin(ω)+F3(ω)Yin(ω),δY(ω)=E4(ω)Xin(ω)+F4(ω)Yin(ω),
E3(ω)=−2κ4G2−u(ω)2[u(ω)+2G cos θ],F3(ω)=−2κ4G2−u(ω)22G sin θ,E4(ω)=−2κ4G2−u(ω)22G sin θ,F4(ω)=−2κ4G2−u(ω)2[u(ω)−2G cos θ].
SX(ω)=[E3(ω)E3(−ω)+F3(ω)F3(−ω)](ncth+12),SY(ω)=[E4(ω)E4(−ω)+F4(ω)F4(−ω)](ncth+12).
⟨δO2⟩=12π∫−∞∞SO(ω)dω,O=X,Y.
σij=⟨fi(t)fj(t)+fj(t)fi(t)⟩/2,i,j=1,2,3,4.
σ˙(t)=Mσ(t)+σ(t)MT+D,
Dij=⟨ni(t)nj(t)+nj(t)ni(t)⟩/2.
Mσ+σMT=−D.
σb=[σb11σb12σb21σb22],
Vb=[e−2rσb11σb12σb21e2rσb22].
W(R)=exp(−12RTVb−1R)2πDet[Vb],
c=1κ2−4G2[iκg′b−2iGg′eiθb†+2Geiθ2κcin†(t)+κ2κcin(t)].
b˙=−κg′2κ2−4G2b+2Gg′2eiθκ2−4G2b†+γmbin(t)+ig′2κκ2−4G2[2Geiθcin†(t)+κcin(t)],
δQ˙=−g′2κ+2GδQ+F1(t)+F2(t),
F1(t)=−ig′κκ+2G[cin†(t)−cin(t)],F2(t)=γm2[bin†(t)+bin(t)],
⟨F1(t1)F1(t2)⟩=g′2κ(κ+2G)2(2ncth+1)δ(t1−t2),⟨F2(t1)F2(t2)⟩=γm2(2nmth+1)δ(t1−t2).
d⟨δQ2(t)⟩dt=−2g′2κ+2G⟨δQ2(t)⟩+g′2κ(κ+2G)2(2ncth+1)+γm2(2nmth+1).
⟨δQ2⟩s=e−2r[κ2(κ+2G)(2ncth+1)+γm(κ+2G)4g′2(2nmth+1)].
ζ=−10 log10⟨δQ2⟩s⟨δQ2⟩vac=−10 log10e−2r−10 log10[κ2(κ+2G)+γm(κ+2G)4g′2]−10 log102,
δZout(ω)=12[δcout(ω)e−iϕ+δcout(−ω)†eiϕ],
δZout(ω)=AZ(ω)Qin(ω)+BZ(ω)Pin(ω)+EZ(ω)Xin(ω)+FZ(ω)Yin(ω),
AZ(ω)=−γm[cos ϕE1(ω)+sin ϕF1(ω)],BZ(ω)=−γm[cos ϕE2(ω)+sin ϕF2(ω)],EZ(ω)=cos ϕH(ω)+sin ϕI(ω),FZ(ω)=cos ϕI(ω)+sin ϕR(ω),H(ω)=2κd(ω)ν(ω){g′2+[u(ω)+2G cos θ]ν(ω)}−1,R(ω)=2κd(ω)ν(ω){g′2+[u(ω)−2G cos θ]ν(ω)}−1,I(ω)=4κd(ω)G sin θν(ω)2.
2πSZout(ω)δ(ω+Ω)=12[⟨δZout(ω)δZout(Ω)⟩+⟨δZout(Ω)δZout(ω)⟩].
SZout(ω)=[AZ(ω)AZ(−ω)+BZ(ω)BZ(−ω)](nmth+12)+[EZ(ω)EZ(−ω)+FZ(ω)FZ(−ω)](ncth+12).

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