Abstract

We discuss ways in which the optomechanical coupling provided by radiation pressure can be used to cool macroscopic collective degrees of freedom such as the vibrational modes of movable mirrors. Cooling is achieved with a phase-sensitive feedback loop that effectively overdamps a mirror’s motion without increasing the thermal noise. The feedback that results can bring macroscopic objects down to the quantum limit. In particular, it is possible to achieve squeezing and entanglement.

© 2003 Optical Society of America

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  2. P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
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  3. M. Pinard, P. F. Cohadon, T. Briant, and A. Heidmann, “Full mechanical characterization of a cold damped mirror,” Phys. Rev. A 63, 013808/1–12 (2000).
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  4. T. Briant, P. F. Cohadon, M. Pinard, and A. Heidmann, “Optical phase-space reconstruction of mirror motion at the attometer level,” Eur. Phys. J. D 22, 131–140 (2003).
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  6. D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401/1–4 (2001).
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  7. D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803/1–19 (2002).
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  9. R. C. Ritter and G. T. Gillies, “Classical limit of mechanical thermal noise reduction by feedback,” Phys. Rev. A 31, 995–1000 (1985), and references therein.
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  10. J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
    [CrossRef]
  11. M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B 280, 555–556 (2000).
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  13. A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301/1–4 (2002).
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  34. F. Haake and R. Reibold, “Strong damping and low-temperature anomalies for the harmonic oscillator,” Phys. Rev. A 32, 2462–2475 (1985).
    [CrossRef] [PubMed]
  35. H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
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  36. F. Grassia, J. M. Courty, S. Reynaud, and P. Touboul, “Quantum theory of fluctuations in a cold damped accelerometer,” Eur. Phys. J. D 8, 101–110 (2000).
    [CrossRef]
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    [CrossRef]
  39. M. Ozawa, “Measurement breaking the standard quantum limit for free-mass position,” Phys. Rev. Lett. 60, 385–388 (1988).
    [CrossRef] [PubMed]
  40. S. M. Tan, “Confirming entanglement in continuous variable quantum teleportation,” Phys. Rev. A 60, 2752–2758 (1999).
    [CrossRef]
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  42. R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
    [CrossRef] [PubMed]
  43. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401/1–4 (2002).
    [CrossRef]
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2003

T. Briant, P. F. Cohadon, M. Pinard, and A. Heidmann, “Optical phase-space reconstruction of mirror motion at the attometer level,” Eur. Phys. J. D 22, 131–140 (2003).
[CrossRef]

2002

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803/1–19 (2002).
[CrossRef]

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301/1–4 (2002).
[CrossRef]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401/1–4 (2002).
[CrossRef]

2001

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401/1–4 (2001).
[CrossRef]

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812/1–8 (2001).
[CrossRef]

R. Folman, J. Schmiedmayer, H. Ritsch, and D. Vitali, “On the observation of decoherence with a movable mirror,” Eur. Phys. J. D 13, 93–107 (2001).
[CrossRef]

2000

M. Pinard, P. F. Cohadon, T. Briant, and A. Heidmann, “Full mechanical characterization of a cold damped mirror,” Phys. Rev. A 63, 013808/1–12 (2000).
[CrossRef]

M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B 280, 555–556 (2000).
[CrossRef]

F. Grassia, J. M. Courty, S. Reynaud, and P. Touboul, “Quantum theory of fluctuations in a cold damped accelerometer,” Eur. Phys. J. D 8, 101–110 (2000).
[CrossRef]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef] [PubMed]

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

1999

S. M. Tan, “Confirming entanglement in continuous variable quantum teleportation,” Phys. Rev. A 60, 2752–2758 (1999).
[CrossRef]

V. Giovannetti, P. Tombesi, and D. Vitali, “Non-Markovian quantum feedback from homodyne measurements: the effect of a nonzero feedback delay time,” Phys. Rev. A 60, 1549–1561 (1999).
[CrossRef]

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller, “Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion,” Phys. Rev. A 60, 538–548 (1999).
[CrossRef]

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999).
[CrossRef]

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

1998

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

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

A. N. Cleland and M. L. Roukes, “A nanometre-scale mechanical electrometer,” Nature 392, 160–162 (1998).
[CrossRef]

1997

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

1995

P. Samphire, R. Loudon, and M. Babiker, “Quantum theory of radiation-pressure fluctuations on a mirror,” Phys. Rev. A 51, 2726–2737 (1995).
[CrossRef] [PubMed]

C. K. Law, “Interaction between a moving mirror and radiation pressure: a Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[CrossRef] [PubMed]

1994

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef] [PubMed]

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

G. J. Milburn, K. Jacobs, and D. F. Walls, “Quantum-limited measurements with the atomic force microscope,” Phys. Rev. A 50, 5256–5263 (1994).
[CrossRef] [PubMed]

H. M. Wiseman, “Quantum theory of continuous feedback,” Phys. Rev. A 49, 2133–2150 (1994).
[CrossRef] [PubMed]

1993

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” Appl. Phys. Lett. 62, 2344–2346 (1993).
[CrossRef]

A. F. Pace, M. J. Collett, and D. F. Walls, “Quantum limits in interferometric detection of gravitational radiation,” Phys. Rev. A 47, 3173–3189 (1993).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

1988

M. Ozawa, “Measurement breaking the standard quantum limit for free-mass position,” Phys. Rev. Lett. 60, 385–388 (1988).
[CrossRef] [PubMed]

1985

F. Haake and R. Reibold, “Strong damping and low-temperature anomalies for the harmonic oscillator,” Phys. Rev. A 32, 2462–2475 (1985).
[CrossRef] [PubMed]

R. C. Ritter and G. T. Gillies, “Classical limit of mechanical thermal noise reduction by feedback,” Phys. Rev. A 31, 995–1000 (1985), and references therein.
[CrossRef] [PubMed]

S. van der Meer, “Stochastic cooling and the accumulation of antiprotons,” Rev. Mod. Phys. 57, 689–697 (1985).
[CrossRef]

1984

H. Grabert, U. Weiss, and P. Talkner, “Quantum theory of the damped harmonic oscillator,” Z. Phys. B 55, 87–94 (1984).
[CrossRef]

1983

H. P. Yuen, “Contractive states and the standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719–722 (1983).
[CrossRef]

1981

R. Loudon, “Quantum limit on the Michelson interferometer used for gravitational-wave detection,” Phys. Rev. Lett. 47, 815–818 (1981).
[CrossRef]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

1980

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
[CrossRef]

Aminoff, C. G.

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999).
[CrossRef]

Armour, A. D.

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301/1–4 (2002).
[CrossRef]

Babiker, M.

P. Samphire, R. Loudon, and M. Babiker, “Quantum theory of radiation-pressure fluctuations on a mirror,” Phys. Rev. A 51, 2726–2737 (1995).
[CrossRef] [PubMed]

Blanc, N.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

Blencowe, M. P.

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301/1–4 (2002).
[CrossRef]

M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B 280, 555–556 (2000).
[CrossRef]

Botkin, D.

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Breitenbach, G.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

Briant, T.

T. Briant, P. F. Cohadon, M. Pinard, and A. Heidmann, “Optical phase-space reconstruction of mirror motion at the attometer level,” Eur. Phys. J. D 22, 131–140 (2003).
[CrossRef]

M. Pinard, P. F. Cohadon, T. Briant, and A. Heidmann, “Full mechanical characterization of a cold damped mirror,” Phys. Rev. A 63, 013808/1–12 (2000).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
[CrossRef]

Cirac, J. I.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef] [PubMed]

Cleland, A. N.

A. N. Cleland and M. L. Roukes, “A nanometre-scale mechanical electrometer,” Nature 392, 160–162 (1998).
[CrossRef]

Cohadon, P. F.

T. Briant, P. F. Cohadon, M. Pinard, and A. Heidmann, “Optical phase-space reconstruction of mirror motion at the attometer level,” Eur. Phys. J. D 22, 131–140 (2003).
[CrossRef]

M. Pinard, P. F. Cohadon, T. Briant, and A. Heidmann, “Full mechanical characterization of a cold damped mirror,” Phys. Rev. A 63, 013808/1–12 (2000).
[CrossRef]

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999).
[CrossRef]

Collett, M. J.

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef] [PubMed]

A. F. Pace, M. J. Collett, and D. F. Walls, “Quantum limits in interferometric detection of gravitational radiation,” Phys. Rev. A 47, 3173–3189 (1993).
[CrossRef] [PubMed]

Conradt, R.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

Courty, J. M.

F. Grassia, J. M. Courty, S. Reynaud, and P. Touboul, “Quantum theory of fluctuations in a cold damped accelerometer,” Eur. Phys. J. D 8, 101–110 (2000).
[CrossRef]

Courty, J.-M.

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[CrossRef]

de Rooij, N. F.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

Duan, L.-M.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef] [PubMed]

Folman, R.

R. Folman, J. Schmiedmayer, H. Ritsch, and D. Vitali, “On the observation of decoherence with a movable mirror,” Eur. Phys. J. D 13, 93–107 (2001).
[CrossRef]

Giedke, G.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef] [PubMed]

Gillies, G. T.

R. C. Ritter and G. T. Gillies, “Classical limit of mechanical thermal noise reduction by feedback,” Phys. Rev. A 31, 995–1000 (1985), and references therein.
[CrossRef] [PubMed]

Giovannetti, V.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401/1–4 (2002).
[CrossRef]

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812/1–8 (2001).
[CrossRef]

V. Giovannetti, P. Tombesi, and D. Vitali, “Non-Markovian quantum feedback from homodyne measurements: the effect of a nonzero feedback delay time,” Phys. Rev. A 60, 1549–1561 (1999).
[CrossRef]

Grabert, H.

H. Grabert, U. Weiss, and P. Talkner, “Quantum theory of the damped harmonic oscillator,” Z. Phys. B 55, 87–94 (1984).
[CrossRef]

Grassia, F.

F. Grassia, J. M. Courty, S. Reynaud, and P. Touboul, “Quantum theory of fluctuations in a cold damped accelerometer,” Eur. Phys. J. D 8, 101–110 (2000).
[CrossRef]

Haake, F.

F. Haake and R. Reibold, “Strong damping and low-temperature anomalies for the harmonic oscillator,” Phys. Rev. A 32, 2462–2475 (1985).
[CrossRef] [PubMed]

Hadjar, Y.

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999).
[CrossRef]

Heidmann, A.

T. Briant, P. F. Cohadon, M. Pinard, and A. Heidmann, “Optical phase-space reconstruction of mirror motion at the attometer level,” Eur. Phys. J. D 22, 131–140 (2003).
[CrossRef]

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[CrossRef]

M. Pinard, P. F. Cohadon, T. Briant, and A. Heidmann, “Full mechanical characterization of a cold damped mirror,” Phys. Rev. A 63, 013808/1–12 (2000).
[CrossRef]

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999).
[CrossRef]

Jacobs, K.

K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller, “Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion,” Phys. Rev. A 60, 538–548 (1999).
[CrossRef]

G. J. Milburn, K. Jacobs, and D. F. Walls, “Quantum-limited measurements with the atomic force microscope,” Phys. Rev. A 50, 5256–5263 (1994).
[CrossRef] [PubMed]

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef] [PubMed]

Kalkbrenner, T.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

Kenny, T. W.

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Kishimoto, Y.

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

Koga, J.

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

Law, C. K.

C. K. Law, “Interaction between a moving mirror and radiation pressure: a Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[CrossRef] [PubMed]

Loudon, R.

P. Samphire, R. Loudon, and M. Babiker, “Quantum theory of radiation-pressure fluctuations on a mirror,” Phys. Rev. A 51, 2726–2737 (1995).
[CrossRef] [PubMed]

R. Loudon, “Quantum limit on the Michelson interferometer used for gravitational-wave detection,” Phys. Rev. Lett. 47, 815–818 (1981).
[CrossRef]

Mancini, S.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401/1–4 (2002).
[CrossRef]

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803/1–19 (2002).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401/1–4 (2001).
[CrossRef]

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

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

Marti, O.

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” Appl. Phys. Lett. 62, 2344–2346 (1993).
[CrossRef]

Mertz, J.

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” Appl. Phys. Lett. 62, 2344–2346 (1993).
[CrossRef]

Milburn, G. J.

G. J. Milburn, K. Jacobs, and D. F. Walls, “Quantum-limited measurements with the atomic force microscope,” Phys. Rev. A 50, 5256–5263 (1994).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

Mlynek, J.

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” Appl. Phys. Lett. 62, 2344–2346 (1993).
[CrossRef]

Müller, T.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

Ozawa, M.

M. Ozawa, “Measurement breaking the standard quantum limit for free-mass position,” Phys. Rev. Lett. 60, 385–388 (1988).
[CrossRef] [PubMed]

Pace, A. F.

A. F. Pace, M. J. Collett, and D. F. Walls, “Quantum limits in interferometric detection of gravitational radiation,” Phys. Rev. A 47, 3173–3189 (1993).
[CrossRef] [PubMed]

Pinard, M.

T. Briant, P. F. Cohadon, M. Pinard, and A. Heidmann, “Optical phase-space reconstruction of mirror motion at the attometer level,” Eur. Phys. J. D 22, 131–140 (2003).
[CrossRef]

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[CrossRef]

M. Pinard, P. F. Cohadon, T. Briant, and A. Heidmann, “Full mechanical characterization of a cold damped mirror,” Phys. Rev. A 63, 013808/1–12 (2000).
[CrossRef]

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999).
[CrossRef]

Raizen, M. G.

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

Reibold, R.

F. Haake and R. Reibold, “Strong damping and low-temperature anomalies for the harmonic oscillator,” Phys. Rev. A 32, 2462–2475 (1985).
[CrossRef] [PubMed]

Reynaud, S.

F. Grassia, J. M. Courty, S. Reynaud, and P. Touboul, “Quantum theory of fluctuations in a cold damped accelerometer,” Eur. Phys. J. D 8, 101–110 (2000).
[CrossRef]

Ribichini, L.

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803/1–19 (2002).
[CrossRef]

Ritsch, H.

R. Folman, J. Schmiedmayer, H. Ritsch, and D. Vitali, “On the observation of decoherence with a movable mirror,” Eur. Phys. J. D 13, 93–107 (2001).
[CrossRef]

Ritter, R. C.

R. C. Ritter and G. T. Gillies, “Classical limit of mechanical thermal noise reduction by feedback,” Phys. Rev. A 31, 995–1000 (1985), and references therein.
[CrossRef] [PubMed]

Roukes, M. L.

A. N. Cleland and M. L. Roukes, “A nanometre-scale mechanical electrometer,” Nature 392, 160–162 (1998).
[CrossRef]

Rugar, D.

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Samphire, P.

P. Samphire, R. Loudon, and M. Babiker, “Quantum theory of radiation-pressure fluctuations on a mirror,” Phys. Rev. A 51, 2726–2737 (1995).
[CrossRef] [PubMed]

Schiller, S.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller, “Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion,” Phys. Rev. A 60, 538–548 (1999).
[CrossRef]

Schmiedmayer, J.

R. Folman, J. Schmiedmayer, H. Ritsch, and D. Vitali, “On the observation of decoherence with a movable mirror,” Eur. Phys. J. D 13, 93–107 (2001).
[CrossRef]

Schwab, K. C.

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301/1–4 (2002).
[CrossRef]

Simon, R.

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

Steinsland, E.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

Stowe, T. D.

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Sundaram, B.

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

Tajima, T.

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

Takuma, H.

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

Talkner, P.

H. Grabert, U. Weiss, and P. Talkner, “Quantum theory of the damped harmonic oscillator,” Z. Phys. B 55, 87–94 (1984).
[CrossRef]

Tan, S. M.

S. M. Tan, “Confirming entanglement in continuous variable quantum teleportation,” Phys. Rev. A 60, 2752–2758 (1999).
[CrossRef]

Tittonen, I.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller, “Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion,” Phys. Rev. A 60, 538–548 (1999).
[CrossRef]

Tombesi, P.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401/1–4 (2002).
[CrossRef]

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803/1–19 (2002).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401/1–4 (2001).
[CrossRef]

V. Giovannetti, P. Tombesi, and D. Vitali, “Non-Markovian quantum feedback from homodyne measurements: the effect of a nonzero feedback delay time,” Phys. Rev. A 60, 1549–1561 (1999).
[CrossRef]

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

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef] [PubMed]

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

Touboul, P.

F. Grassia, J. M. Courty, S. Reynaud, and P. Touboul, “Quantum theory of fluctuations in a cold damped accelerometer,” Eur. Phys. J. D 8, 101–110 (2000).
[CrossRef]

van der Meer, S.

S. van der Meer, “Stochastic cooling and the accumulation of antiprotons,” Rev. Mod. Phys. 57, 689–697 (1985).
[CrossRef]

Vitali, D.

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803/1–19 (2002).
[CrossRef]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401/1–4 (2002).
[CrossRef]

R. Folman, J. Schmiedmayer, H. Ritsch, and D. Vitali, “On the observation of decoherence with a movable mirror,” Eur. Phys. J. D 13, 93–107 (2001).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401/1–4 (2001).
[CrossRef]

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812/1–8 (2001).
[CrossRef]

V. Giovannetti, P. Tombesi, and D. Vitali, “Non-Markovian quantum feedback from homodyne measurements: the effect of a nonzero feedback delay time,” Phys. Rev. A 60, 1549–1561 (1999).
[CrossRef]

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

Wago, K.

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Walls, D. F.

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef] [PubMed]

G. J. Milburn, K. Jacobs, and D. F. Walls, “Quantum-limited measurements with the atomic force microscope,” Phys. Rev. A 50, 5256–5263 (1994).
[CrossRef] [PubMed]

A. F. Pace, M. J. Collett, and D. F. Walls, “Quantum limits in interferometric detection of gravitational radiation,” Phys. Rev. A 47, 3173–3189 (1993).
[CrossRef] [PubMed]

Weiss, U.

H. Grabert, U. Weiss, and P. Talkner, “Quantum theory of the damped harmonic oscillator,” Z. Phys. B 55, 87–94 (1984).
[CrossRef]

Wiseman, H. M.

K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller, “Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion,” Phys. Rev. A 60, 538–548 (1999).
[CrossRef]

H. M. Wiseman, “Quantum theory of continuous feedback,” Phys. Rev. A 49, 2133–2150 (1994).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

Wybourne, M. N.

M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B 280, 555–556 (2000).
[CrossRef]

Yasamura, K.

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Yuen, H. P.

H. P. Yuen, “Contractive states and the standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719–722 (1983).
[CrossRef]

Zoller, P.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett.

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” Appl. Phys. Lett. 62, 2344–2346 (1993).
[CrossRef]

T. D. Stowe, K. Yasamura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Eur. Phys. J. D

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

F. Grassia, J. M. Courty, S. Reynaud, and P. Touboul, “Quantum theory of fluctuations in a cold damped accelerometer,” Eur. Phys. J. D 8, 101–110 (2000).
[CrossRef]

R. Folman, J. Schmiedmayer, H. Ritsch, and D. Vitali, “On the observation of decoherence with a movable mirror,” Eur. Phys. J. D 13, 93–107 (2001).
[CrossRef]

T. Briant, P. F. Cohadon, M. Pinard, and A. Heidmann, “Optical phase-space reconstruction of mirror motion at the attometer level,” Eur. Phys. J. D 22, 131–140 (2003).
[CrossRef]

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[CrossRef]

Europhys. Lett.

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999).
[CrossRef]

Nature

A. N. Cleland and M. L. Roukes, “A nanometre-scale mechanical electrometer,” Nature 392, 160–162 (1998).
[CrossRef]

Phys. Rev. A

H. M. Wiseman, “Quantum theory of continuous feedback,” Phys. Rev. A 49, 2133–2150 (1994).
[CrossRef] [PubMed]

V. Giovannetti, P. Tombesi, and D. Vitali, “Non-Markovian quantum feedback from homodyne measurements: the effect of a nonzero feedback delay time,” Phys. Rev. A 60, 1549–1561 (1999).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401/1–4 (2001).
[CrossRef]

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803/1–19 (2002).
[CrossRef]

M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima, “Stochastic cooling of atoms using lasers,” Phys. Rev. A 58, 4757–4760 (1998).
[CrossRef]

R. C. Ritter and G. T. Gillies, “Classical limit of mechanical thermal noise reduction by feedback,” Phys. Rev. A 31, 995–1000 (1985), and references therein.
[CrossRef] [PubMed]

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: towards observation of quantum limits,” Phys. Rev. A 59, 1038–1044 (1999).
[CrossRef]

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812/1–8 (2001).
[CrossRef]

C. K. Law, “Interaction between a moving mirror and radiation pressure: a Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
[CrossRef] [PubMed]

A. F. Pace, M. J. Collett, and D. F. Walls, “Quantum limits in interferometric detection of gravitational radiation,” Phys. Rev. A 47, 3173–3189 (1993).
[CrossRef] [PubMed]

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef] [PubMed]

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

K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller, “Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion,” Phys. Rev. A 60, 538–548 (1999).
[CrossRef]

G. J. Milburn, K. Jacobs, and D. F. Walls, “Quantum-limited measurements with the atomic force microscope,” Phys. Rev. A 50, 5256–5263 (1994).
[CrossRef] [PubMed]

M. Pinard, P. F. Cohadon, T. Briant, and A. Heidmann, “Full mechanical characterization of a cold damped mirror,” Phys. Rev. A 63, 013808/1–12 (2000).
[CrossRef]

S. M. Tan, “Confirming entanglement in continuous variable quantum teleportation,” Phys. Rev. A 60, 2752–2758 (1999).
[CrossRef]

F. Haake and R. Reibold, “Strong damping and low-temperature anomalies for the harmonic oscillator,” Phys. Rev. A 32, 2462–2475 (1985).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

P. Samphire, R. Loudon, and M. Babiker, “Quantum theory of radiation-pressure fluctuations on a mirror,” Phys. Rev. A 51, 2726–2737 (1995).
[CrossRef] [PubMed]

Phys. Rev. D

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Phys. Rev. Lett.

H. P. Yuen, “Contractive states and the standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719–722 (1983).
[CrossRef]

M. Ozawa, “Measurement breaking the standard quantum limit for free-mass position,” Phys. Rev. Lett. 60, 385–388 (1988).
[CrossRef] [PubMed]

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

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
[CrossRef]

R. Loudon, “Quantum limit on the Michelson interferometer used for gravitational-wave detection,” Phys. Rev. Lett. 47, 815–818 (1981).
[CrossRef]

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301/1–4 (2002).
[CrossRef]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef] [PubMed]

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401/1–4 (2002).
[CrossRef]

Physica B

M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B 280, 555–556 (2000).
[CrossRef]

Rev. Mod. Phys.

S. van der Meer, “Stochastic cooling and the accumulation of antiprotons,” Rev. Mod. Phys. 57, 689–697 (1985).
[CrossRef]

Z. Phys. B

H. Grabert, U. Weiss, and P. Talkner, “Quantum theory of the damped harmonic oscillator,” Z. Phys. B 55, 87–94 (1984).
[CrossRef]

Other

C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991).

M. D. Reid, “The Einstein–Podolsky–Rosen paradox and entanglement. 1. Signatures of EPR correlations for continuous variables,” arXiv.org e-Print archive, URL http://arXiv.org/abs/quant-ph/0112038.

S. Mancini, D. Vitali, V. Giovannetti, and P. Tombesi, “Stationary entanglement between macroscopic mechanical oscillators,” arXiv.org e-Print archive, URL http://arXiv.org/abs/quant-ph/0209014.

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Figures (8)

Fig. 1
Fig. 1

Schematic description of a linear Fabry–Perot cavity with end oscillating mirror M. The equilibrium cavity length is L. A cavity mode is driven by an input laser beam. The output field is subjected to homodyne detection (D). The signal is then fed back to the mirror motion (dashed curve).

Fig. 2
Fig. 2

Schematic description of a ring cavity with two oscillating mirrors, M1 and M2. The equilibrium distance between them is L (this is also the distance between the fixed mirrors). A cavity mode is driven by an input laser beam. The output field is subjected to homodyne detection (D). The signal is then fed back to the mirror motion (dashed curve).

Fig. 3
Fig. 3

Rescaled steady-state energy 2Ust/ωm versus rescaled input power ζ, plotted for several values of g2 (a, g2=10; b, g2=103; c, g2=105; d, g2=107), with kBT/ωm=105 and η=0.8. The optimal input power increases correspondingly, and, for high gain values, ground-state cooling can be obtained.

Fig. 4
Fig. 4

Rescaled steady-state energy 2Ust/ωm versus rescaled input power ζ, plotted for several values of g1 (a, g1=10; b, g1=103; c, g1=105; d, g1=107) at fixed Q=107, and with kBT/ωm=105 and η=0.8. The optimal input power ζopt increases correspondingly, and, for high gain values, ground-state cooling can be achieved.

Fig. 5
Fig. 5

Rescaled steady-state energy 2Ust/ωm versus ζ for increasing values of mechanical quality factor Q (a, Q=103; b, Q=105; c, Q=107) at fixed g1=107, and with kBT/ωm=105 and η=0.8.

Fig. 6
Fig. 6

Steady state position–momentum correlation -QP+PQst versus ζ for three values of the feedback gain: a, g1=105; b, g1=106; c, g1=107. The other parameters are Q=104, kBT/ωm=105, and η=0.8.

Fig. 7
Fig. 7

Steady-state position variance Q2st versus ζ for two values of the feedback gain, g1=107 (short-dashed curve) and g1=109 (solid curve). The long-dashed line denotes the standard quantum limit Q2st=1/4. The other parameters are Q=104, kBT/ωm=105, and η=0.8.

Fig. 8
Fig. 8

Marker of entanglement E versus feedback gain g3 for three values of the mechanical quality factor: a, Q=103; b, Q=3×103; c, Q=104. The other parameters are kBT/ωm=105 and η=0.8.

Equations (81)

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H=ωcbb+ωm(P2+Q2)-2GbbQ+iE[bexp(-iω0t)-b exp(iω0t)],
Q˙(t)=ωmP(t),
P˙(t)=-ωmQ(t)+W(t)-γmP(t)+Gb(t)b(t),
b˙(t)=-[iωc-iω0+(γc/2)]b(t)+2iGQ(t)b(t)+E+γcbin(t),
bin(t)bin(t)=bin(t)bin(t)=0,
bin(t)bin(t)=δ(t-t).
W(t)W(t)=12πγmωm {Fr(t-t)+iFi(t-t)},
Fr(t)=0ω¯dωω cos(ωt)cothω2kBT,
Fi(t)=-0ω¯dωω sin(ωt),
Q˙(t)=ωmP(t),
P˙(t)=-ωmQ(t)-γmP(t)+2GβX(t)+W(t),
Y˙(t)=-γc2 Y(t)+2GβQ(t)+γc2 Yin(t),
X˙(t)=-γc2 X(t)+γc2 Xin(t),
Q˙1(t)=ωmP1(t),
P˙1(t)=-ωmQ1(t)+W1(t)-γmP1(t)+G˜b(t)b(t),
Q˙2(t)=ωmP2(t),
P˙2(t)=-ωmQ2(t)+W2(t)-γmP2(t)-G˜b(t)b(t),
b˙(t)=-iωc-iω0+γc2b(t)+2iG˜[Q1(t)-Q2(t)]b(t)+E+γcbin(t),
Q˙+(t)=ωmP+(t),
P˙+(t)=-ωmQ+(t)+W+(t)-γmP+(t),
Q˙-(t)=ωmP-(t),
P˙-(t)=-ωmQ-(t)+W-(t)-γmP-(t)+2G˜b(t)b(t),
b˙(t)=-iωc-iω0+γc2b(t)+22iG˜Q-(t)b(t)+E+γcbin(t),
Y(t)4Gβγc Q(t)+Yin(t)γc,
Yout(t)=2ηγcY(t)-ηYinη(t),
bη(t)bη(t)=bη(t)bη(t)=0,
bη(t)bη(t)=δ(t-t),
bin(t)bη(t)=bη(t)bin(t)=ηδ(t-t).
O˙fb(t)=i γcη0tdtGmf(t)Yout(t-t)[gmfP(t), O(t)],
Q˙(t)=ωmP(t)+gmfγcY(t)-gmf2γcη Yinη(t),
P˙(t)=-ωmQ(t)-γmP(t)+2GβX(t)+W(t),
Y˙(t)=-γc2 Y(t)+2GβQ(t)+γc2 Yin(t),
X˙(t)=-γc2 X(t)+γc2 Xin(t),
ζ=16G2β2γmγc=64G2ω0γmγc2 
Q˙(t)=ωmP(t)-γmg1Q(t)-γmζ g1Yin(t)+γmηζg12 Yinη(t),
P˙(t)=-ωmQ(t)-γmP(t)+½γmζXin(t)+W(t).
O˙fb(t)=iηγc0tdtGcd(t)Yout(t-t)[gcdQ(t), O(t)].
Q˙(t)=ωmP(t),
P˙(t)=-ωmQ(t)-γmP(t)+2GβX(t)-gcdY˙(t)+gcd2γcη Y˙inη(t)+W(t),
Y˙(t)=-γc2 Y(t)+2GβQ(t)+γc2 Yin(t),
X˙(t)=-γc2 X(t)+γc2 Xin(t).
Q˙(t)=ωmP(t),
P˙(t)=-ωmQ(t)-γmP(t)+12γmζXin(t)+W(t)-γmg2ωm Q˙(t)-g2γmωmζ Y˙in(t)+g2γm2ωmηζ Y˙inη(t).
Y˙in(t)Y˙in(t)=Y˙in(t)Y˙in(t)=Y˙inη(t)Y˙inη(t)=Y˙inη(t)Y˙inη(t)=-δ¨(t-t),
Y˙inη(t)Y˙in(t)=Y˙in(t)Y˙inη(t)=-ηδ¨(t-t),
Xin(t)Y˙inη(t)=-Y˙inη(t)Xin(t)=-iηδ˙(t-t),
Q¨(t)+(1+g1)γmQ˙(t)+(ωm2+γm2g1)Q(t)
=ωm12γmζXin(t)+W(t)-γmζ g1[Y˙in(t)+γmYin(t)]+γmηζg12 [Y˙inη(t)+γmYinη(t)]
Q¨(t)+(1+g2)γmQ˙(t)+ωm2Q(t)
=ωm12γmζXin(t)+W(t)-g2γmωmζ Y˙in(t)
+g2γm2ωmηζ Y˙inη(t)
Q2st=γmdω2π |χ˜cd(ω)|2ζ4+g224ηζ|G˜cd(ω)|2ωm2+ω2ωmcothω2kBTΘ[-ω¯,ω¯](ω),
P2st=γm-dω2πω2ωm2 |χ˜cd(ω)|2ζ4+g224ηζ|G˜cd(ω)|2ωm2+ω2ωmcothω2kBTΘ[-ω¯,ω¯](ω),
χ˜cd(ω)=ωmωm2-ω2+iωγm(1+g2)
Q2stba=P2stba=ζ8(1+g2).
Q2stfb=P2stfb=g228ηζ(1+g2).
Q2stBM=kBT2ωm(1+g2).
P2stBM=kBT2ωm11+g2+γmπωmlnω¯2πkT.
Q2st=P2st=g228ηζ+ζ8+kBT2ωm11+g2.
PQ+QPst=1ωmlimtddt Q(t)2=0,
Ust=2ωmQ2st=ωm4(1+g2)g22ηζ+ζ+4kBTωm.
Ust=ωm2g21+g21η+2kBTωm1g2,
Q2stba=ζQ28(1+g1)(Q2+g1),
P2stba=ζ(Q2+g12+g1)8(1+g1)(Q2+g1)
Q2stfb=g128ηζ1+Q2+g1(1+g1)(Q2+g1),
P2stfb=g128ηζQ2(1+g1)(Q2+g1)
Q2stBM=kBT2ωmQ2(1+g2)(Q2+g1).
P2BM=kBT2ωmg12+Q2+g1(1+g1)(Q2+g1)+γmπωmlnω¯2πkT.
Q2st=g128ηζ1+Q2+g1(1+g1)(Q2+g1)+ζ8+kBT2ωmQ2(1+g1)(Q2+g1),
P2st=g128ηζQ2(1+g1)(Q2+g1)+ζ8+kBT2ωmg12+Q2+g1(1+g1)(Q2+g1)+γmπωmlnω¯2πkT.
Ust=ωm[Q2st+P2st]=ωm8g12ηζ(1+2Q2+g1)(1+g1)(Q2+g1)+ζ+4kBTωm(g12+2Q2+g1)(1+g1)(Q2+g1).
Ustωm21η+2kBTωm1g1,
QP+PQst2=ζ8+kBT2ωmg1Q(1+g1)(Q2+g1)-g128ηζQ(1+g1)(Q2+g1).
ϕ=12arctanQP+PQst(Q2st-P2st)
Q2stmin=g1Q(1+Q2+g1)1/24η(1+g1)(Q2+g1)+kBT2ωmQ2(1+g1)(Q2+g1).
O˙fb(t)=i γcη0tdtGmf-(t)Yout(t-t)×[gmf-P-(t), O(t)],
ζ˜=32G˜2β˜2γmγc=128G˜2ω0γmγc2 ,
Q-2st=g328ηζ˜1+Q2+g3(1+g3)(Q2+g3)+ζ˜8+kBT2ωmQ2(1+g3)(Q2+g3).
E=16Q-2stP+2st<1,
P+2BM=kBT2ωm+γmπωmlnω¯2πkT.
ζ˜opt=g3Q1+Q2+g3η1/2,

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