Abstract

Quantum squeezing, as a typical quantum effect, is an important resource for many applications in quantum technologies. Here we propose a scheme for generating quantum squeezing, including the ponderomotive squeezing and the mechanical squeezing, in an optomechanical system, in which the radiation-pressure coupling and the mechanical spring constant are modulated periodically. In this system, the radiation-pressure interaction can be enhanced remarkably by the modulation-induced mechanical parametric amplification. Moreover, the effective phonon noise can be suppressed completely by introducing a squeezed vacuum reservoir. This ultimately leads to that our scheme can achieve a controllable quantum squeezing. Numerical calculations show that our scheme is experimentally realizable with current technologies.

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

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  48. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
    [Crossref] [PubMed]

2017 (1)

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu, “Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95(5), 053861 (2017).
[Crossref]

2016 (1)

2015 (1)

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed Optomechanics with Phase-Matched Amplification and Dissipation,” Phys. Rev. Lett. 114(9), 093602 (2015).
[Crossref] [PubMed]

2014 (3)

A. Nunnenkamp, V. Sudhir, A. K. Feofanov, A. Roulet, and T. J. Kippenberg, “Quantum-limited amplification and parametric instability in the reversed dissipation regime of cavity optomechanics,” Phys. Rev. Lett. 113(2), 023604 (2014).
[Crossref] [PubMed]

E. A. Sete and H. Eleuch, “Strong squeezing and robust entanglement in cavity electromechanics,” Phys. Rev. A 89(1), 013841 (2014).
[Crossref]

Y. Xiao, Y. F. Yu, and Z. M. Zhang, “Controllable optomechanically induced transparency and ponderomotive squeezing in an optomechanical system assisted by an atomic ensemble,” Opt. express 22(15), 17979–17989 (2014).
[Crossref] [PubMed]

2013 (5)

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(1), 013835 (2013).
[Crossref]

Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref] [PubMed]

U. B. Hoff, G. I. Harris, L. S. Madsen, H. Kerdoncuff, M. Lassen, B. M. Nielsen, W. P. Bowen, and U. L. Andersen, “Quantum-enhanced micromechanical displacement sensitivity,” Opt. Lett. 38(9), 1413–1415 (2013).
[Crossref] [PubMed]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7(3), 229 (2013).
[Crossref]

A. Szorkovszky, G. A. Brawley, A. C. Doherty, and W. P. Bowen, “Strong thermomechanical squeezing via weak measurement,” Phys. Rev. Lett. 110(18), 184301 (2013).
[Crossref] [PubMed]

2012 (2)

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

D. W. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488(7412), 476–480 (2012).
[Crossref] [PubMed]

2011 (7)

E. A. Sete, S. Das, and H. Eleuch, “External-field effect on quantum features of radiation emitted by a quantum well in a microcavity,” Phys. Rev. A 83(2), 023822 (2011).
[Crossref]

The LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys.  7(12), 962 (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(21), 213603 (2011).
[Crossref] [PubMed]

Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L. S. Ma, and C. W. Oates, “Making optical atomic clocks more stable with 10-16-level laser stabilization,” Nat. Photonics 5(3), 158 (2011).
[Crossref]

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425 (2011).
[Crossref]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

J. D. Teufel, D. Li, 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(7337), 204–208 (2011).
[Crossref] [PubMed]

2010 (4)

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104(25), 251102 (2010).
[Crossref] [PubMed]

E. A. Sete and H. Eleuch, “Interaction of a quantum well with squeezed light: Quantum-statistical properties,” Phys. Rev. A 82(4), 043810 (2010).
[Crossref]

F. Marino, F. S. Cataliotti, A. Farsi, M. S. de Cumis, and F. Marin, “Classical signature of ponderomotive squeezing in a suspended mirror resonator,” Phys. Rev. Lett. 104(7), 073601 (2010).
[Crossref] [PubMed]

F. Marino, F. S. Cataliotti, A. Farsi, M. S. de Cumis, and F. Marin, “Classical signature of ponderomotive squeezing in a suspended mirror resonator,” Phys. Rev. Lett. 104(7), 073601 (2010).
[Crossref] [PubMed]

2009 (2)

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(6), 063819 (2009).
[Crossref]

J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79(5), 052102 (2009).
[Crossref]

2008 (1)

H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A 78(1), 013824 (2008).
[Crossref]

2007 (3)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99(9), 093901 (2007).
[Crossref] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical Duffing resonator,” Phys. Rev. Lett. 98(7), 078103 (2007).
[Crossref] [PubMed]

2006 (1)

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
[Crossref] [PubMed]

2004 (1)

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70(20), 205304 (2004).
[Crossref]

2003 (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[Crossref] [PubMed]

2002 (1)

E. Giacobino, J. P. Karr, G. Messin, H. Eleuch, and A. Baas, “Quantum optical effects in semiconductor microcavities,” C. R. Physique 3(1), 41–52 (2002).
[Crossref]

1999 (2)

H. Eleuch, J. M. Courty, G. Messin, C. Fabre, and E. Giacobino, “Cavity QED effects in semiconductor microcavities,” J. Opt. B: Quantum Semiclass. Opt. 1(1), 1–7 (1999).
[Crossref]

G. Messin, J. P. Karr, H. Eleuch, J. M. Courty, and E. Giacobino, “Squeezed states and the quantum noise of light in semiconductor microcavities,” J. Phys.: Condens. Matter 11(31), 6069–6078 (1999).

1994 (2)

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49(2), 1337–1343 (1994).
[Crossref] [PubMed]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49(5), 4055–4065 (1994).
[Crossref] [PubMed]

1991 (1)

D. Rugar and P. Grutter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67(6), 699–702 (1991).
[Crossref] [PubMed]

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(12), 5288–5290 (1987).
[Crossref]

1985 (1)

M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32(5), 2887–2892 (1985).
[Crossref]

1982 (1)

H. Paul, “Photon anti-bunching,” Rev. Mod. Phys. 54(4), 1061–1102 (1982).
[Crossref]

1980 (1)

R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Progr. Phys. 43(7), 913 (1980).
[Crossref]

1970 (1)

V. B. Braginskii, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 31, 829 (1970).

1967 (1)

V. B. Braginskii and A. B. Manukin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 25(4), 653–655 (1967).

Alegre, T. P.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Allman, M. S.

J. D. Teufel, D. Li, 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(7337), 204–208 (2011).
[Crossref] [PubMed]

Almog, R.

R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical Duffing resonator,” Phys. Rev. Lett. 98(7), 078103 (2007).
[Crossref] [PubMed]

Andersen, U. L.

Arcizet, O.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
[Crossref] [PubMed]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[Crossref] [PubMed]

Aspelmeyer, M.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Baas, A.

E. Giacobino, J. P. Karr, G. Messin, H. Eleuch, and A. Baas, “Quantum optical effects in semiconductor microcavities,” C. R. Physique 3(1), 41–52 (2002).
[Crossref]

Bachor, H. A.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7(3), 229 (2013).
[Crossref]

Bauchrowitz, J.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104(25), 251102 (2010).
[Crossref] [PubMed]

Bergquist, J. C.

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425 (2011).
[Crossref]

Bohm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Botter, T.

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Muller-Ebhardt, H.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104(25), 251102 (2010).
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Nooshi, N.

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T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104(25), 251102 (2010).
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A. Nunnenkamp, V. Sudhir, A. K. Feofanov, A. Roulet, and T. J. Kippenberg, “Quantum-limited amplification and parametric instability in the reversed dissipation regime of cavity optomechanics,” Phys. Rev. Lett. 113(2), 023604 (2014).
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T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “Quantum enhancement of the zero-area Sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104(25), 251102 (2010).
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J. D. Teufel, D. Li, 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(7337), 204–208 (2011).
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I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99(9), 093901 (2007).
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T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu, “Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95(5), 053861 (2017).
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R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, “Noise squeezing in a nanomechanical Duffing resonator,” Phys. Rev. Lett. 98(7), 078103 (2007).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
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Zheng, L. L.

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu, “Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95(5), 053861 (2017).
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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(6), 063819 (2009).
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C. R. Physique (1)

E. Giacobino, J. P. Karr, G. Messin, H. Eleuch, and A. Baas, “Quantum optical effects in semiconductor microcavities,” C. R. Physique 3(1), 41–52 (2002).
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J. Opt. B: Quantum Semiclass. Opt. (1)

H. Eleuch, J. M. Courty, G. Messin, C. Fabre, and E. Giacobino, “Cavity QED effects in semiconductor microcavities,” J. Opt. B: Quantum Semiclass. Opt. 1(1), 1–7 (1999).
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J. Phys.: Condens. Matter (1)

G. Messin, J. P. Karr, H. Eleuch, J. M. Courty, and E. Giacobino, “Squeezed states and the quantum noise of light in semiconductor microcavities,” J. Phys.: Condens. Matter 11(31), 6069–6078 (1999).

Nat. Photonics (3)

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7(3), 229 (2013).
[Crossref]

Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L. S. Ma, and C. W. Oates, “Making optical atomic clocks more stable with 10-16-level laser stabilization,” Nat. Photonics 5(3), 158 (2011).
[Crossref]

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425 (2011).
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Nat. Phys (1)

The LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys.  7(12), 962 (2011).
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Nature (6)

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

D. W. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488(7412), 476–480 (2012).
[Crossref] [PubMed]

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

J. D. Teufel, D. Li, 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(7337), 204–208 (2011).
[Crossref] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[Crossref] [PubMed]

Opt. express (2)

Opt. Lett. (1)

Phys. Rev. A (12)

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu, “Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95(5), 053861 (2017).
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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(1), 013835 (2013).
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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(6), 063819 (2009).
[Crossref]

J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79(5), 052102 (2009).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Sketch of the system. A cavity mode pumped by a laser at the red-detuned mechanical sideband, couples to a mechanical resonator with a modulating coupling strength g(t) = g0 cos (ωdt), in addition, the mechanical resonator is parametrically driven by modulating its spring constant, i.e., k(t) = k0kr cos(2ωdt).
Fig. 2
Fig. 2 The effective thermal noise Neff is plotted as a function of (a) the squeezing parameters re, (b) the reference phase Φ e for different values of the amplitude ϵb, where ϵb = 3999.50κ (solid black curve), ϵb = 3999.65κ (dashed blue curve), ϵb = 3999.80κ (dotted red curve), ϵb = 3999.95κ (dot-dashed green curve). Other parameters are ∆ m = 4000κ and κ = 0.1MHz.
Fig. 3
Fig. 3 The steady-state amplitudes |cs| and |bs| vs pump amplitude ϵp. The parameters are κ = 0.1MHz, γm = 0.01κ, g0 = 0.005κ, ∆ m = 4000κ, ϵb = 3999.95κ, Δ c = ω m .
Fig. 4
Fig. 4 The intensity spectrum of the transmitted field is plotted as a function of normalized frequency ( ω ω c ) / ω m for different values of the amplitude ϵb. Other parameters are the same as in Fig. 3.
Fig. 5
Fig. 5 The squeezing spectrum of the transmitted field is plotted as a function of normalized frequency ( ω ω c ) / ω m for different values of the amplitude ϵb. Other parameters are the same as in Fig. 3.
Fig. 6
Fig. 6 The squeezing spectrum is plotted as a function of (a) the squeezing parameter re, (b) the reference phase Φ e . Other parameters are the same as in Fig. 3.
Fig. 7
Fig. 7 The steady-state variance Δ X 2 s s is plotted as a function of (a) the squeezing parameter re, (b) the reference phase Φ e . Other parameters are the same as in Fig. 3.

Equations (60)

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H t = Δ c c c + Δ m b b + 1 2 ϵ b ( b 2 + b 2 ) 1 2 g 0 c c ( b + b ) + i ϵ p ( c c ) ,
S ( r d ) b S ( r d ) = b cosh ( r d ) + b sinh ( r d ) ,
H e f f = S ( r d ) H t S ( r d ) = Δ c c c + ω m b b g c c ( b + b ) + i ϵ p ( c c ) ,
c ˙ = ( κ + i Δ c ) c + i g c ( b + b ) + ϵ p + 2 κ c i n ,
b ˙ = ( γ m + i ω m ) b + i g c c + 2 γ m b i n ,
c i n ( t ) c i n ( t ) = δ ( t t ) ,
b i n ( t ) b i n ( t ) = ( N e f f + 1 ) δ ( t t ) ,
b i n ( t ) b i n ( t ) = N e f f δ ( t t ) ,
b i n ( t ) b i n ( t ) = M e f f δ ( t t ) ,
b i n ( t ) b i n ( t ) = M e f f δ ( t t ) ,
N e f f = sinh 2 ( r e ) cosh 2 ( r d ) + sinh 2 ( r d ) cosh 2 ( r e ) + 1 2 cos ( Φ e ) sinh ( 2 r e ) sinh ( 2 r d ) ,
M e f f = [ cosh ( r e ) sinh ( r d ) + e i Φ e sinh ( r e ) cosh ( r d ) ] × [ cosh ( r e ) cosh ( r d ) + e i Φ e sinh ( r e ) sinh ( r d ) ] .
c s = ϵ p i Δ + κ ,
b s = g | c s | 2 ω m i γ m ,
δ c ˙ = ( κ + i Δ ) δ c + i g c s ( δ b + δ b ) + 2 κ c i n ,
δ b ˙ = ( γ m + i ω m ) δ b + i g ( c s * δ c + c s δ c ) 2 γ m b i n .
μ ˙ ( t ) = A μ ( t ) + ν ( t ) ,
A = ( ( κ + i Δ ) 0 i g c s i g c s 0 ( κ i Δ ) i g c s * i g c s * i g c s * i g c s ( γ m + i ω m ) 0 i g c s * i g c s 0 ( γ m i ω m ) ) .
4 γ m κ { [ ( γ m + κ ) 2 + Δ 2 ] 2 + 2 [ ( γ m + κ ) 2 Δ 2 ] ω m 2 + ω m 4 } + 16 ( γ m + κ ) 2 Δ ω m g 2 | c s | 2 > 0 ,
( Δ 2 + κ 2 ) ( γ m 2 + ω m 2 ) 4 Δ ω m g 2 | c s | 2 > 0 ,
o ( ω ) = 1 2 π o ( t ) e i ω t d t , o ( ω ) = [ o ( ω ) ] ,
c i n ( ω ) c i n ( ω ) = δ ( ω + ω ) ,
b i n ( ω ) b i n ( ω ) = ( N e f f + 1 ) δ ( ω + ω ) ,
b i n ( ω ) b i n ( ω ) = N e f f δ ( ω + ω )
b i n ( ω ) b i n ( ω ) = M e f f * δ ( ω + ω ) ,
b i n ( ω ) b i n ( ω ) = M e f f δ ( ω + ω ) .
δ c ( ω ) = ζ 1 ( ω ) c i n + ζ 2 ( ω ) c i n + ζ 3 ( ω ) b i n + ζ 4 ( ω ) b i n ,
ζ 1 ( ω ) = 2 κ [ Θ 1 ( g 2 | c s | 2 + Θ 2 Λ 2 ) Θ 2 g 2 | c s | 2 ] d ( ω )
ζ 2 ( ω ) = 2 κ ( Θ 1 Θ 2 ) g 2 c s 2 d ( ω ) ,
ζ 3 ( ω ) = i 2 γ m Θ 2 Λ 2 g c s d ( ω )
ζ 4 ( ω ) = i 2 γ m Θ 1 Λ 2 g c s d ( ω )
d ( ω ) = Θ 1 [ g 2 | c s | 2 ( Λ 1 Λ 2 ) + Θ 2 Λ 1 Λ 2 ] + Θ 2 g 2 | c s | 2 ( Λ 2 Λ 1 ) ,
S o u t ( ω ) = δ c o u t ( ω ) δ c o u t ( ω ) ,
S θ ( ω ) = δ X θ o u t ( ω ) δ X θ o u t ( ω ) ,
S θ ( ω ) = 1 + 2 B c c ( ω ) + [ e 2 i θ B c c ( ω ) + c . c ] ,
δ c o u t ( ω ) δ c o u t ( ω ) = δ ( ω + ω ) B c c ( ω ) ,
δ c o u t ( ω ) δ c o u t ( ω ) = δ ( ω + ω ) B c c ( ω ) .
e 2 i θ o p t = ± B c c ( ω ) | B c c ( ω ) | .
S o p t ( ω ) = 1 + 2 B c c ( ω ) 2 | B c c ( ω ) | ,
B c c ( ω ) = 2 κ [ | ζ 2 ( ω ) | 2 + | ζ 3 ( ω ) | 2 N e f f + ζ 3 * ( ω ) ζ 4 ( ω ) M e f f + ζ 4 * ( ω ) ζ 3 ( ω ) M e f f * + | ζ 4 ( ω ) | 2 ( N e f f + 1 ) ] ,
B c c ( ω ) = 2 κ [ ζ 1 ( ω ) ζ 2 ( ω ) + ζ 3 ( ω ) ζ 3 ( ω ) M e f f * + ζ 3 ( ω ) ζ 4 ( ω ) ( N e f f + 1 ) + ζ 4 ( ω ) ζ 3 ( ω ) N e f f + ζ 4 ( ω ) ζ 4 ( ω ) M e f f ζ 2 ( ω ) 2 κ ] .
B c c ( ω ) = 2 κ [ | ζ 2 ( ω ) | 2 + | ζ 4 ( ω ) | 2 ] ,
B c c ( ω ) = 2 κ [ ζ 1 ( ω ) ζ 2 ( ω ) + ζ 3 ( ω ) ζ 4 ( ω ) ζ 2 ( ω ) 2 κ ] ,
ω 1 = ± z 1 z 1 2 + z 2 2 ,
ω 2 = ± z 1 + z 1 2 + z 2 2 ,
z 1 = γ 2 + ω m 2 + κ 2 + Δ 2 + 4 γ κ ,
z 2 = 16 Δ ω m + g 2 | c s | 2 4 ( Δ 2 + κ 2 ) ( γ 2 + ω m 2 ) .
Δ X ± 2 = [ 1 + 2 δ b δ b ± ( δ b 2 + δ b 2 ) ( δ b ± δ b ) 2 ] e ± 2 r d .
H e f f , L = Δ δ c δ c + ω m δ b δ b g c s δ c δ b g c s * δ c δ b .
δ c ˙ = ( κ + i Δ ) δ c + i g c s δ b + 2 κ c i n ,
δ b ˙ = ( γ m + i ω m ) δ b + i g c s * δ c + 2 γ m b i n .
δ b ( t ) = f 1 ( t ) δ c ( 0 ) + 2 κ 0 t f 1 ( t t ) c i n ( t ) d t + f 2 ( t ) δ b ( 0 ) + 2 γ m 0 t f 2 ( t t ) b i n ( t ) d t ,
f 1 ( t ) = i g c s * [ e ( u v ) t / 2 e ( u + v ) t / 2 ] u ,
f 2 ( t ) = [ u + v 2 ( γ m + i ω m ) ] e ( u ν ) t / 2 + [ u + v 2 ( κ + i Δ ) ] e ( u + v ) t / 2 2 u ,
u = [ γ m κ i ( Δ ω m ) ] 2 4 g 2 | c s | 2 ,
v = γ m + κ + i ( Δ + ω m ) .
δ c ( 0 ) δ c ( 0 ) = 0 , δ b ( 0 ) δ b ( 0 ) = 0.
Δ X + 2 s s = [ 1 + 4 γ m N e f f η 1 ± 2 γ m M e f f * η 2 ± 2 γ m M e f f η 2 * ] e ± 2 r d ,
η 1 = κ [ ( γ m + κ ) 2 + ( Δ ω m ) 2 ] + ( γ m + κ ) g 2 | c s | 2 2 γ m κ [ ( γ m + κ ) 2 + ( Δ ω m ) 2 ] + 2 ( γ m + κ ) 2 g 2 | c s | 2 ,
η 2 = ( κ + i Δ ) [ γ m + κ + i ( Δ + ω m ) ] + g 2 | c s | 2 2 [ ( κ + i Δ ) ( γ m + ω m ) + g 2 | c s | 2 ] [ γ m + κ + i ( Δ + ω m ) ] .

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