Abstract

Optically trapped nanoparticles have recently emerged as exciting candidates for tests of quantum mechanics at the macroscale and as versatile platforms for ultrasensitive metrology. Recent experiments have demonstrated parametric feedback cooling, nonequilibrium physics, and temperature detection, all in the classical regime. Here we provide the first quantum model for trapped nanoparticle cooling and force sensing. In contrast to existing theories, our work indicates that the nanomechanical ground state may be prepared without using an optical resonator; that the cooling mechanism corresponds to nonlinear friction; and that the energy loss during cooling is nonexponential in time. Our results show excellent agreement with experimental data in the classical limit, and constitute an underlying theoretical framework for experiments aiming at ground state preparation. Our theory also addresses the optimization of, and the fundamental quantum limit to, force sensing, thus providing theoretical direction to ongoing searches for ultraweak forces using levitated nanoparticles.

© 2016 Optical Society of America

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References

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  1. L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
    [Crossref]
  2. J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
    [Crossref]
  3. J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
    [Crossref]
  4. J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
    [Crossref]
  5. L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).
  6. T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
    [Crossref]
  7. Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
    [Crossref]
  8. J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).
  9. Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
    [Crossref]
  10. M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
    [Crossref]
  11. J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
    [Crossref]
  12. L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).
  13. J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
    [Crossref]
  14. N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
    [Crossref]
  15. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
    [Crossref]
  16. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
    [Crossref]
  17. O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
    [Crossref]
  18. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  19. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. Lpz. 525, 215–233 (2013).
    [Crossref]
  20. P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).
  21. A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
    [Crossref]
  22. G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
    [Crossref]
  23. D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
    [Crossref]
  24. A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
    [Crossref]
  25. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 2002).
  26. L. Diósi, “Quantum master equation of a particle in a gas environment,” Europhys. Lett. 30, 63–68 (1995).
    [Crossref]
  27. D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
    [Crossref]
  28. C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed. (Springer, 2004).
  29. S. Mancini, D. Vitali, and P. Tombesi, “Optomechanical cooling of a macroscopic oscillator by homodyne feedback,” Phys. Rev. Lett. 80, 688–691 (1998).
    [Crossref]
  30. H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
    [Crossref]
  31. 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, 093901 (2007).
    [Crossref]
  32. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
    [Crossref]
  33. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
    [Crossref]
  34. O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
    [Crossref]
  35. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).
  36. P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
    [Crossref]
  37. N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” J. Phys. A. 25, 5677–5691 (1992).
    [Crossref]
  38. J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995).
    [Crossref]
  39. K. Hornberger, Entanglement and Decoherence, Vol. 768 of Lecture Notes in Physics (Springer, 2009).
  40. T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
    [Crossref]
  41. L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
    [Crossref]

2015 (6)

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

2014 (5)

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
[Crossref]

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

2013 (8)

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

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

2012 (2)

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

2011 (4)

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[Crossref]

2010 (2)

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
[Crossref]

2009 (1)

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

2008 (1)

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (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, 093901 (2007).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

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

1998 (1)

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

1995 (2)

L. Diósi, “Quantum master equation of a particle in a gas environment,” Europhys. Lett. 30, 63–68 (1995).
[Crossref]

J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995).
[Crossref]

1993 (1)

H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
[Crossref]

1992 (1)

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

Anders, J.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Arita, Y.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

Arndt, M.

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Asenbaum, P.

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Aspelmeyer, M.

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

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Atherton, D. P.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Barker, P.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Barker, P. F.

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

Bateman, J.

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

Berthelot, J.

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

Blaser, F.

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

Bose, S.

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 2002).

Chang, D. E.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Cirac, J. I.

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

Clerk, A. A.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Cross, M. C.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Cunningham, M.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Deesuwan, T.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Delic, U.

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

Dellago, C.

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

Deutsch, B.

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

Dholakia, K.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

Diósi, L.

L. Diósi, “Quantum master equation of a particle in a gas environment,” Europhys. Lett. 30, 63–68 (1995).
[Crossref]

Fonseca, P. Z. G.

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed. (Springer, 2004).

Genes, C.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Geraci, A. A.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
[Crossref]

Gerry, C.

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

Gieseler, J.

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

Gigan, S.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Girvin, S. M.

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

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Gisin, N.

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

Grass, D.

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

Gratta, G.

D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
[Crossref]

Halliwell, J.

J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995).
[Crossref]

Hornberger, K.

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

K. Hornberger, Entanglement and Decoherence, Vol. 768 of Lecture Notes in Physics (Springer, 2009).

Juan, M. L.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

Kaltenbaek, R.

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

Karabalin, R. B.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Kenig, E.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Kheifets, S.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[Crossref]

Kiesel, N.

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

Kim, M. S.

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

Kimble, H. J.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

Kippenberg, T. J.

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

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, 093901 (2007).
[Crossref]

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

Kitching, J.

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
[Crossref]

Knight, P.

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

Kuhn, S.

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Li, T.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[Crossref]

Li, T. C.

Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

Mancini, S.

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

Marquardt, F.

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

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

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Matheny, M. H.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Mavrogordatos, T.

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

Mazilu, M.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

Mestres, P.

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

Meystre, P.

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

Milburn, G. J.

H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
[Crossref]

Millen, J.

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Monteiro, T. S.

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

Moore, D. C.

D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
[Crossref]

Moritz, C.

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

Morley, G. W.

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

Neukirch, L. P.

L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

Nick Vamivakas, A.

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

Nimmrichter, S.

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Nooshi, N.

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, 093901 (2007).
[Crossref]

Novotny, L.

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

Painter, O.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

Papp, S. B.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
[Crossref]

Percival, I. C.

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

Peterson, R. W.

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

Pflanzer, A. C.

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

Purdy, T. P.

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

Quidant, R.

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. Nick Vamivakas, “Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,” Opt. Lett. 38, 2976–2979 (2013).
[Crossref]

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

Raizen, M. G.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[Crossref]

Ranjit, G.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Regal, C. A.

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

Rider, A. D.

D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
[Crossref]

Romero-Isart, O.

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

Rosenholm, J. M.

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

Roukes, M. L.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Scala, M.

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

Sezer, U.

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Spasenovic, M.

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

Stutz, J. H.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

Tombesi, P.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

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

Ulbricht, H.

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

Vahala, K. J.

Vamivakas, A. N.

L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

Villanueva, L. G.

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Vitali, D.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

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

von Haartman, E.

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

Wilson, D. J.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

Wilson-Rae, I.

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, 093901 (2007).
[Crossref]

Wiseman, H. M.

H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
[Crossref]

Ye, J.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

Yin, Z. Q.

Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

Zoller, P.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed. (Springer, 2004).

Zoupas, A.

J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995).
[Crossref]

Zwerger, W.

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, 093901 (2007).
[Crossref]

Ann. Phys. Lpz. (1)

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

Appl. Phys. Lett. (1)

P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, “Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,” Appl. Phys. Lett. 107, 151102 (2015).
[Crossref]

Contemp. Phys. (1)

L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56, 48–62 (2015).

Europhys. Lett. (1)

L. Diósi, “Quantum master equation of a particle in a gas environment,” Europhys. Lett. 30, 63–68 (1995).
[Crossref]

Int. J. Mod. Phys. B (1)

Z. Q. Yin, A. A. Geraci, and T. C. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

J. Phys. A. (1)

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

Nano Lett. (1)

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, E. Kenig, M. C. Cross, and M. L. Roukes, “A nanoscale parametric feedback oscillator,” Nano Lett. 11, 5054–5059 (2011).
[Crossref]

Nat. Commun. (3)

J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, “Near-field interferometry of a free-falling nanoparticle from a point-like source,” Nat. Commun. 5, 4788 (2014).

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, “Cavity cooling of free silicon nanoparticles in high vacuum,” Nat. Commun. 4, 2743 (2013).

Nat. Nanotechnol. (1)

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Nat. Photonics (1)

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. Nick Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photonics 9, 653–657 (2015).

Nat. Phys. (1)

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[Crossref]

New J. Phys. (1)

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, “Non-equilibrium steady state of a driven levitated particle with feedback cooling,” New J. Phys. 17, 045011 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (4)

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91, 051805 (2015).
[Crossref]

A. C. Pflanzer, O. Romero-Isart, and J. I. Cirac, “Master-equation approach to optomechanics with arbitrary dielectrics,” Phys. Rev. A 86, 013802 (2012).
[Crossref]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[Crossref]

Phys. Rev. D (1)

J. Halliwell and A. Zoupas, “Quantum state diffusion, density matrix diagonalization, and decoherent histories: a model,” Phys. Rev. D 52, 7294–7307 (1995).
[Crossref]

Phys. Rev. Lett. (11)

J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, “Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,” Phys. Rev. Lett. 112, 103603 (2014).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref]

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, “Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).
[Crossref]

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

H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback via homodyne detection,” Phys. Rev. Lett. 70, 548–551 (1993).
[Crossref]

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, 093901 (2007).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014).
[Crossref]

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105, 101101 (2010).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref]

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref]

Physics (1)

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

Proc. Natl. Acad. Sci. USA (2)

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010).
[Crossref]

N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

Rev. Mod. Phys. (1)

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

Science (1)

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

Other (4)

K. Hornberger, Entanglement and Decoherence, Vol. 768 of Lecture Notes in Physics (Springer, 2009).

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 2002).

C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed. (Springer, 2004).

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. (a) Image of the trapped nanoparticle and (b) schematic of the experiment modeled in this article. The schematic shows the electro-optic modulator (EOM), polarizing beam splitters (PBS1 and PBS2), high numerical-aperture lenses (L1 and L2), the detector (D), the beam dump (BD), and the feedback circuit gain (G).
Fig. 2.
Fig. 2. (a) Diffusive evolution of the trapped nanoparticle’s position at atmospheric pressure, (b) harmonic motion of the nanoparticle at a lower pressure of 4 × 10 3 mbar. The reduction in the amplitude of the harmonic motion corresponds to the turning on of feedback, i.e., cooling.
Fig. 3.
Fig. 3. (a)  y and z phonon cooling dynamics [Eq. (13)], (b) steady-state phonon number versus pressure [Eq. (14a)]. The circles represent the experimental data, and the solid curve is our theoretical model for a fused silica sphere ( ϵ r = 2.1 and density = 2200    kg / m 3 ) of radius r d = 50    nm , 1064 nm trap (100 mW) and probe (10 mW) beams, mechanical frequency ω z / 2 π = 38    kHz , χ 10 7 , and trap intensity modulation M 0.1 % . The dotted lines represent the equivalent curves for one of the transverse degrees of freedom ( ω y / 2 π = 138    kHz ). The dashed curve in (a) represents the prediction of our theory for a setup placed in a cryostat with the feedback chosen optimally, keeping M 10 % .
Fig. 4.
Fig. 4. Experimentally measured positional PSDs for all three degrees of freedom, with the dark lines representing the theoretical fits to the data [Eq. (18)]. Data were taken at a moderate vacuum pressure of 10 mbars and clearly show the Lorentzian shape of the resonance. These fits were used to extract the values of ω j , Γ , and N ss .
Fig. 5.
Fig. 5. (a) Plots of the shot noise force PSD [the last term of Eq. (19)] versus the normalized mechanical frequency ω / ω z for low and high total damping Γ . The minimum occurs for ω opt = ω z 2 Γ 2 / 2 . (b) Plot of the force sensitivity as a function of the normalized optical power Φ / Φ SQL at high vacuum. The standard quantum limit is reached when the shot noise balances the recoil and backaction noise [Eq. (19)].

Equations (21)

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H = H m + H f + H int .
H f = ω p a a + μ d 3 k ω k a μ ( k ) a μ ( k ) ,
H = H S + H B + H S B ,
H S = ω p a a + j ω j b j b j j g j a a ( b j + b j ) ,
ρ ˙ ( t ) = 1 i [ H S , ρ ] A t 2 D [ Q ] ρ + L sc ρ ,
B [ ρ ( t ) ] = D p 2 D [ Q z ] ρ D q 2 D [ P z ] ρ i η f 4 m [ Q z , { P z , ρ } ] ,
a out = a in + α χ 2 Q z ( t ) ,
I h = χ 2 Φ Q z ( t ) + χ 2 Φ ξ ( t ) ,
M Δ I t I t G χ 2 Φ b z b z ω z ,
F [ ρ ( t ) ] = i χ 2 Φ G [ Q z 3 , { P z , ρ } ] χ 2 Φ G 2 D [ Q z 3 ] ρ ,
ρ ˙ ( t ) = 1 i [ H ˜ S , ρ ( t ) ] ( A t + A p ) D [ Q z ] ρ ( t ) / 2 B D [ a ] ρ ( t ) + B [ ρ ( t ) ] + F [ ρ ( t ) ] ,
N ˙ = 2 J N 2 ( J + K ) N + L .
N ( t ) = ( J + K ) 4 J + 1 2 J τ tanh ( t τ + θ ) ,
N ss lim t N ( t ) = 1 2 J τ ( J + K ) 4 J
η f 2 m N 0 2 J = D p + A t + A p 2 J ,
Q ˙ z = L 0 # [ Q z ] = ω z P z , P ˙ z = L 0 # [ P z ] + F / m ω z z = ω z Q z Γ P z + F / m ω z z ,
S T = 2 m Γ 0 k B T eff , S F = 54 m ω z χ 2 Φ G 2 ( 2 N 2 + 2 N + 1 ) .
χ m ( ω ) = { m [ ( ω z 2 ω 2 ) i ω Γ ] } 1
| q ˜ z ( ω ) | 2 = | χ m | 2 ( S T + S F ) + z 2 χ 2 Φ ,
| F ˜ ( ω ) | 2 = S T + S F + S S ( ω ) ,
| F ˜ | 2 SQL = 2 m Γ 0 k B T + 4 m ω z ( A t + 2 Γ ) .

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