Abstract

Signal reception of nuclear magnetic resonance (NMR) usually relies on electrical amplification of the electromotive force caused by nuclear induction. Here, we report up-conversion of a radio-frequency NMR signal to an optical regime using a high-stress silicon nitride membrane that interfaces the electrical detection circuit and an optical cavity through the electro-mechanical and the opto-mechanical couplings. This enables optical NMR detection without sacrificing the versatility of the traditional nuclear induction approach. While the signal-to-noise ratio is currently limited by the Brownian motion of the membrane as well as additional technical noise, we find it can exceed that of the conventional electrical schemes by increasing the electro-mechanical coupling strength. The electro-mechano-optical NMR detection presented here opens the possibility of mechanical parametric amplification of NMR signals. Moreover, it can potentially be combined with the laser cooling technique applied to nuclear spins.

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

Full Article  |  PDF Article
OSA Recommended Articles
Optical nuclear magnetic resonance: theory, simulation, and animation

Myron W. Evans and Chris R. Pelkie
J. Opt. Soc. Am. B 9(7) 1020-1029 (1992)

Nuclear magnetic resonance imaging

William P. Rothwell
Appl. Opt. 24(23) 3958-3968 (1985)

References

  • View by:
  • |
  • |
  • |

  1. T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
    [Crossref]
  2. F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
    [Crossref]
  3. E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
    [Crossref]
  4. E. Purcell, “Nuclear magnetism in relation to problems of the liquid and solid states,” Science 107, 433–440 (1948).
    [Crossref]
  5. F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
    [Crossref]
  6. D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
    [Crossref]
  7. A. Abragam, Principles of Nuclear Magnetism (Oxford University, 1961).
  8. C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
    [Crossref]
  9. J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
    [Crossref]
  10. I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
    [Crossref]
  11. M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
    [Crossref]
  12. H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
    [Crossref]
  13. T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
    [Crossref]
  14. I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
    [Crossref]
  15. E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 580–594 (1950).
    [Crossref]
  16. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
    [Crossref]
  17. J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
    [Crossref]
  18. C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
    [Crossref]
  19. C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
    [Crossref]
  20. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
    [Crossref]
  21. E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.
  22. C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
    [Crossref]
  23. C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
    [Crossref]
  24. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
    [Crossref]
  25. O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
    [Crossref]
  26. A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
    [Crossref]
  27. L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
    [Crossref]
  28. D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
    [Crossref]
  29. Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
    [Crossref]
  30. E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
    [Crossref]
  31. A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
    [Crossref]
  32. C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
    [Crossref]
  33. M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
    [Crossref]
  34. M. Mehring, Principles of High-Resolution NMR in Solids, 2nd ed. (Springer, 1983).
  35. K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
    [Crossref]
  36. K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
    [Crossref]

2017 (1)

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

2016 (2)

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
[Crossref]

2015 (1)

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

2014 (3)

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
[Crossref]

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

2013 (3)

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

2012 (2)

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

2011 (4)

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
[Crossref]

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

2010 (4)

M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

2006 (4)

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

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

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
[Crossref]

2005 (1)

I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
[Crossref]

2000 (1)

J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
[Crossref]

1997 (1)

D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
[Crossref]

1993 (1)

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

1950 (1)

E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 580–594 (1950).
[Crossref]

1948 (1)

E. Purcell, “Nuclear magnetism in relation to problems of the liquid and solid states,” Science 107, 433–440 (1948).
[Crossref]

1946 (3)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
[Crossref]

E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

Abe, E.

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

Abragam, A.

A. Abragam, Principles of Nuclear Magnetism (Oxford University, 1961).

Appel, J.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Arcizet, O.

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

Ardavan, A.

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

Aspelmeyer, M.

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

Auffeves, A.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Awschalom, D. D.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
[Crossref]

Bagci, T.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Barthe, M. F.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Bäuerle, D.

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

Becerra, L. R.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Bergonzo, P.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Bertet, P.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Bhakar, B.

D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
[Crossref]

Bienfait, A.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Blaser, F.

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

Bloch, F.

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
[Crossref]

Böhm, H. R.

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

Borneman, T. W.

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

Briant, T.

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

Briggs, G. A. D.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Buckley, B. B.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Butler, M. C.

M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
[Crossref]

Cho, S. U.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Clerk, A. A.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

Cohadon, P. F.

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

Cory, D. G.

C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
[Crossref]

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

Degen, C. L.

M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
[Crossref]

Del’Haye, P.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Devoret, M. H.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

DiCarlo, L.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Dodd, S.

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

Dréau, A.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Du, J.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Eichler, C.

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

Esteve, D.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Frunzio, L.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Gerfen, G. J.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Gigan, S.

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

Ginossar, E.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Girvin, S. M.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

Griffin, R. G.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Hahn, E. L.

E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 580–594 (1950).
[Crossref]

Hakonen, P.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Hansen, W.

F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
[Crossref]

Heidmann, A.

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

Heikkilä, T. T.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Hertzberg, J. B.

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

Hoult, D. I.

D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
[Crossref]

Ichijo, N.

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

Jacques, V.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Jelezko, F.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Kikkawa, J. M.

J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
[Crossref]

Kim, M.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

Kippenberg, T. J.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Kitagawa, K.

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

Koretsky, A.

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

Kubo, Y.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Langer, G.

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

Lee, S.-K.

I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
[Crossref]

Lo, C. C.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Lyon, S. A.

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

Mamin, H. J.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

Marcus, C. M.

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

Marquardt, F.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

Massel, F.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Mehring, M.

M. Mehring, Principles of High-Resolution NMR in Solids, 2nd ed. (Springer, 1983).

Meijer, J.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Meriles, C. A.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Morton, J. J. L.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Murphy-Boesch, J.

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

Noda, Y.

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

Nooshi, N.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Ohno, K.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

Ong, F. R.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Packard, M.

F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
[Crossref]

Paternostro, M.

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

Petta, J. R.

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

Pezzagna, S.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Pinard, M.

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

Pirkkalainen, J.-M.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Pla, J. J.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Poggio, M.

M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
[Crossref]

Polzik, E. S.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

Pound, R.

E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

Purcell, E.

E. Purcell, “Nuclear magnetism in relation to problems of the liquid and solid states,” Science 107, 433–440 (1948).
[Crossref]

E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

Qian, C.

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

Reinhard, F.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Rettner, C. T.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

Roch, J.-F.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Romalis, M. V.

I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
[Crossref]

I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
[Crossref]

Rugar, D.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

Saloniemi, H.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Savukov, I. M.

I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
[Crossref]

I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
[Crossref]

Schenkel, T.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Schliesser, A.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

Schmid, S.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Schoelkopf, R. J.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Schuster, D. I.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Schwab, K. C.

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

Sears, A. P.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Sherwood, M. H.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

Shi, F.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Sigillito, A. J.

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

Sillanpää, M. A.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Simonsen, A.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Singel, D. J.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Slichter, C. P.

C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
[Crossref]

Sørensen, A.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Sørensen, A. S.

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

Staudacher, T.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Stern, M.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Takahashi, M.

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

Takeda, K.

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

Takegoshi, K.

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

Taylor, J. M.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

Temkin, R. J.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Torrey, H.

E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

Usami, K.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Vahala, K. J.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Villanueva, L. G.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Vion, D.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Weis, C. D.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Weitekamp, D. P.

M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
[Crossref]

Wood, C. J.

C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
[Crossref]

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

Wrachtrup, J.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Wu, H.

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Yamada, K.

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

Zabow, G.

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

Zeilinger, A.

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

Zeuthen, E.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

Zheng, D.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Zhou, X.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Appl. Phys. Lett. (1)

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

Chem. Phys. Lett. (1)

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

Concepts Magn. Reson. (1)

D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
[Crossref]

J. Magn. Reson. (2)

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

Magn. Reson. Med. (1)

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

Nanotechnology (1)

M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
[Crossref]

Nature (6)

I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
[Crossref]

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

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

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

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Phys. Rev. (4)

F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
[Crossref]

E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 580–594 (1950).
[Crossref]

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

Phys. Rev. A (2)

C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
[Crossref]

M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
[Crossref]

Phys. Rev. Lett. (8)

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
[Crossref]

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

Rep. Prog. Phys. (1)

C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
[Crossref]

Rev. Mod. Phys. (1)

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

Science (4)

J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
[Crossref]

E. Purcell, “Nuclear magnetism in relation to problems of the liquid and solid states,” Science 107, 433–440 (1948).
[Crossref]

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Other (3)

A. Abragam, Principles of Nuclear Magnetism (Oxford University, 1961).

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

M. Mehring, Principles of High-Resolution NMR in Solids, 2nd ed. (Springer, 1983).

Supplementary Material (1)

NameDescription
» Supplement 1       Supplementary Material

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. (a) Experimental setup for EMO NMR composed of an orthogonal pair of coils tuned at the NMR frequency, a membrane put inside a vacuum chamber, an optical cavity, and a photodetector. (b) Schematic drawing of the membrane capacitor. The Au layer on the membrane is electrically floating, and coupled capacitively to the Al pattern on the substrate. The two electrodes of the capacitor were electrically connected with the rest of the circuit through a pair of contact probes pushing against the Al pads on the silica substrate. (c) Photograph of the Au-deposited membrane.
Fig. 2.
Fig. 2. Drive-power dependence of the sideband spectra of the optically-detected membrane oscillation under application of a continuous-wave tone signal with a power of 81    dBm . The spectra are plotted with vertical offsets proportional to the drive power. The baselines (horizontal broken lines) indicate the corresponding drive power (right axis) as well as the reference power spectral density of 114.5    dBm / Hz . Along with the membrane spectra (blue lines) the peaks corresponding to the tone signals (red lines) appear at 500-Hz off-resonance from the mechanical resonance frequency ω m (black points). The observed downward shifts of the mechanical resonance frequency were fitted with a model discussed in Supplement 1 (orange line).
Fig. 3.
Fig. 3. H 1 spin echo signals in a 0.1 mol/L aqueous solution of CuSO 4 detected by the EMO approach on-resonance (blue line) and + 2.5    kHz off-resonance (red line). The vertical scale represents the 5000-times average signal intensity in units of the number of photons reaching the photodetector per second. The broken line represents a convolution of the electrically detected spin-echo signal shown in the inset with an exponential function with a time constant 2 / γ m . The signal-to-noise ratio S / N is about 5.4.
Fig. 4.
Fig. 4. Schematic diagram of electro-mechano-optical signal transduction of NMR. The three harmonic oscillators—the LC circuit, the membrane, and the optical cavity—are represented with circles, each of which has channels of the input and output, with coupling strengths κ i , G em , G om , and κ o , and dissipation to the bath, with rates γ i , γ m and γ o . The RF signal S generated by nuclear induction at frequency ω s ω LC is transduced to the membrane oscillation through the LC circuit with the electro-mechanical coupling under application of the drive signal at ω D = ω LC + ω m . The resultant membrane oscillation is in turn read out optically with the optical cavity through the opto-mechanical coupling.

Tables (1)

Tables Icon

Table 1. Noise Budget of the Current EMO NMR Detection

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

q ˙ = Δ i ϕ κ iT 2 q κ i ( Q in + S ) γ i q in ,
ϕ ˙ = Δ i q κ iT 2 ϕ κ i ϕ in γ i ϕ in G em z ,
z ˙ = ω m p ,
p ˙ = ω m z γ m p 2 γ m f in G em q G om X ,
X ˙ = Δ o Y κ oT 2 X κ o X in ,
Y ˙ = Δ o X κ oT 2 Y κ o Y in G om z ,
X out = X in + κ o X .
z ¨ ω m = ω m z γ m ω m z ˙ 2 γ m f in ( G em q + G om X ) .
z ( ω ) = χ m ( ω ) [ 2 γ m f in ( ω ) ( G em q ( ω ) + G om X ( ω ) ) ] ,
χ m ( ω ) = ( ω 2 ω m i ω γ m ω m + ω m ) 1 .
q ( ω ) = χ LC ( ω ) [ Δ i G em z + κ i ( Δ i ϕ in + ( i ω κ iT 2 ) Q in ) + γ i ( Δ i ϕ in + ( i ω κ iT 2 ) q in ) ] ,
X ( ω ) = χ c ( ω ) [ Δ o G om z + κ o ( Δ o Y in + ( i ω κ oT 2 ) X in ) ] ,
χ LC ( ω ) = [ ( i ω + κ iT 2 ) 2 + Δ i 2 ] 1 ,
χ c ( ω ) = [ ( i ω + κ oT 2 ) 2 + Δ o 2 ] 1 .
X out = κ o κ oT Y in + ( 1 κ o κ oT ) X in G om κ o κ oT 2 γ m χ m ( ω ) f in G om κ o κ oT G em i ω κ iT 2 χ m ( ω ) [ κ i ( Q in + S ) + γ i q in ] .
X ˜ out = X out cos ( Ω t ) + Y out sin ( Ω t ) + κ o N D cos ( Ω D t ) ,
Y out = κ o κ oT X in + ( 1 κ o κ oT ) Y in + G om κ o κ oT 2 γ m χ m ( ω ) f in + G om κ o κ oT G em i ω κ iT 2 χ m ( ω ) [ κ i ( Q in + S ) + γ i q in ] .
O ( ω ) = κ o N D | X out | 2 + | Y out | 2 .
S F F ( ω ) = n th ( ω m , T eff ) ,
S q q ( ω ) = n th ( ω LC , T ) ,
n th ( ω , T ) = k B T ω .
S oo ( ω ) = κ o N D [ ( ( κ o κ oT ) 2 + ( 1 κ o κ oT ) 2 ) ( 2 S X X ( ω ) + 2 S Y Y ( ω ) ) + C om κ o κ oT 2 γ m 2 | χ m ( ω ) | 2 4 S F F ( ω ) + C om κ o κ oT C em ( ω ) γ m 2 | χ m ( ω ) | 2 × [ 4 S q q ( ω ) + κ i κ iT 4    S 2 δ ( ω ω m ) ] ] .
C om = G om 2 γ m κ oT ,
C em ( ω ) = 4 G em 2 γ m κ iT κ iT 2 4 ω 2 + κ iT 2 .
S 2 κ iT κ i ( S X X ( ω m ) + S Y Y ( ω m ) 2 C om κ o κ oT C em ( ω m ) + 2 S F F ( ω m ) C em ( ω m ) + S q q ( ω m ) ) Δ ,
L ( ω ) = δ P ω 2 + δ P 2 4 ,
L ( ω ) P D ω D ,
S N = S 2 T 2 * 2 ( γ m T 2 * 2 ) κ iT κ i S X X ( ω m ) + S Y Y ( ω m ) 2 C om κ o κ oT C em ( ω m ) + κ iT κ i 2 S F F ( ω m ) C em ( ω m ) + κ iT κ i S q q ( ω m ) + η p P D ω D γ m 0.12
η p = ω m Δ 2 ω m + Δ 2 d ω 2 π γ m 2 | χ m ( ω ) | 2 L ( ω ) ,

Metrics