Abstract

We analyze magnetometry using an optically levitated nanodiamond. We consider a configuration where a magnetic field gradient couples the mechanical oscillation of the diamond with its spin degree of freedom provided by a nitrogen vacancy center. First, we investigate the measurement of the position spectrum of the mechanical oscillator. We find that conditions of ultrahigh vacuum and feedback cooling allow a magnetic field gradient sensitivity of 1μTm1/Hz. At high pressure and room temperature, this sensitivity degrades and can attain a value of the order of 100mTm1/Hz. Subsequently, we characterize the magnetic field gradient sensitivity obtainable by maneuvering the spin degrees of freedom using Ramsey interferometry. We find that this technique can offer photon-shot noise and spin-projection noise limited magnetic field gradient sensitivity of 100μTm1/Hz. We conclude that this hybrid levitated nanomechanical magnetometer provides a favorable and versatile platform for sensing applications.

© 2017 Optical Society of America

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2017 (2)

R. M. Pettit, L. P. Neukirch, Y. Zhang, and A. N. Vamivakas, “Coherent control of a single nitrogen-vacancy center spin in optically levitated nanodiamond,” J. Opt. Soc. Am. B 34(6), C31–C35 (2017).
[Crossref]

T. Delord, L. Nicolas, L. Schwab, and G. Hétet, “Electron spin resonance from NV centers in diamonds levitating in an ion trap,” New. J. Phys. 19, 033031 (2017).
[Crossref]

2016 (11)

W. Ge, B. Rodenburg, and M. Bhattacharya, “Feedback-induced bistability of an optically levitated nanoparticle: A Fokker-Planck treatment,” Phys. Rev. A 94(2), 023808 (2016).
[Crossref]

T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Qi Yin, and T. Li, “Torsional optomechanics of a levitated nonspherical nanoparticle,” Phys. Rev. Lett. 117(12), 123604 (2016).
[Crossref] [PubMed]

K. Xia and J. Twamley, “Generating spin squeezing states and Greenberger-Horne-Zeilinger entanglement using hybrid phonon-spin ensemble in diamond,” Phys. Rev. B 94(20), 205118 (2016).
[Crossref]

C. Yu, J. Janousek, E. Sheridan, D. L. McAuslan, H. R. Dunlop, P. K. Lam, Y. Zhang, and W. P. Bowen, “Optomechanical magnetometry with a macroscopic resonator,” Phys. Rev. App. 5(4), 044007 (2016).
[Crossref]

C. Wan, M. Scala, S. Bose, A. C. Frangeskou, ATM A. Rahman, G. W. Morley, P. F. Barker, and M. S. Kim, “Tolerence in the Ramsey interference of a trapped nanodiamond,” Phys. Rev. A 93(4), 043852 (2016).
[Crossref]

Q. Hou, W. Yang, C. Chen, and Z. Yin, “Generation of macroscopic Schrödinger cat states in diamond mechanical resonator,” Sci. Rep. 6, 37542 (2016).
[Crossref]

T. M. Hoang, J. Ahn, J. Bang, and T. Li, “Electron spin control of optically levitated nanodiamonds in vacuum,” Nat. Commun. 7, 12250 (2016).
[Crossref] [PubMed]

D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical quantum control of nitrogen-vacancy center in diamond,” Phys. Rev. Lett. 116(14), 143602 (2016).
[Crossref] [PubMed]

W. Ge and M. Bhattacharya, “Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence,” New J. Phys. 18, 103002 (2016).
[Crossref]

B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, “Quantum model of cooling and force sensing with an optically trapped nanoparticle,” Optica 3(3), 318 (2016).
[Crossref]

V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, “Direct measurement of photon recoil from a levitated nanoparticle,” Phys. Rev. Lett. 116(24), 243601 (2016).
[Crossref] [PubMed]

2015 (2)

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using micro-spheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91(5), 051805 (2015).
[Crossref]

L. P. Neukirch, E. Hartmaan, J. M. Rosenholm, and A. N. Vamivakas, “Multi-dimensional single-spin nano-optomechanics with levitated nanodiamond,” Nat. Phton. 9, 653 (2015).
[Crossref]

2014 (8)

A. Albrecht, A. Retzker, and M. B. Plenio, “Testing quantum gravity by nanodiamond interferometry with nitrogen-vacancy centers,” Phys. Rev. A 90(3), 033834 (2014).
[Crossref]

P. Ovartchaiyapong, K. W. Lee, B. A. Myers, and A. C. B. Jayich, “Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator,” Nat. Commun. 5, 4429 (2014).
[Crossref] [PubMed]

D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113(25), 251801 (2014).
[Crossref]

L. P. Neukirch and A. N. Vamivakas, “Nano-optomechanics with optically levitated nanoparticles,” Contemp. Phys. 56(1), 48–62 (2014).

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

L. Rondin, J. -P. Tetienne, T. Hingant, J. -F. Roch, P. Maletinsky, and V. Jacques, “Magnetometry with nitrogen-vacancy defects in diamond,” Rep. Prog. Phys. 77, 056503 (2014).
[Crossref] [PubMed]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90(4), 043848 (2014).
[Crossref]

S. Forstner, E. Sheridan, J. Knittel, C. L. Humphreys, G. A. Brawley, H. R. Dunlop, and W. P. Bowen, “Ultrasensitive optomechanical magnetometry,” Adv. Mater. 26(36), 6348–6353 (2014).
[Crossref] [PubMed]

2013 (7)

K. Fang, V. M. Acosta, C. Santori, Z. Huang, K. M. Itoh, H. Watanabe, S. Shikata, and R. G. Beausoleil, “High-sensitivity magnetometry based on quantum beats in diamond nitrogen-vacancy centers,” Phys. Rev. Lett. 110(13), 130802 (2013).
[Crossref] [PubMed]

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

Z. Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27(26), 1330018 (2013).
[Crossref]

J. Q. Zhang, S. Zhang, J. H. Zou, L. Chen, W. Yang, Y. Li, and M. Feng, “Fast optical cooling of nanomechanical cantilever with the dynamic zeeman effect,” Opt. Express 21(24), 29695–29710 (2013).
[Crossref]

S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, “Phonon-induced spin-spin interactions in diamond nanostructures: Application to spin squeezing,” Phys. Rev. Lett. 110(15), 156402 (2013).
[Crossref] [PubMed]

Z. Yin, T. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88(3), 033614 (2013).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111(18), 180403 (2013).
[Crossref] [PubMed]

2012 (5)

S. Kolkowitz, A. C. B. Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. G. E. Harris, and M. D. Lukin, “Coherent sensing of a mechanical resonator with a single-spin qubit,” Science 335(6076), 1603–1606 (2012).
[Crossref] [PubMed]

O. Romero-Isart, L. Clemente, C. Navau, A. Sanchez, and J. I. Cirac, “Quantum magnetomechanics with levitating superconducting microspheres,” Phys. Rev. Lett. 109(14), 147205 (2012).
[Crossref] [PubMed]

M. Cirio, G. K. Brennen, and J. Twamley, “Quantum magnetomechanics: Ultrahigh-Q-levitated mechanical oscillators,” Phys. Rev. Lett. 109(14), 147206 (2012).
[Crossref] [PubMed]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525(3), 215–233 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swain, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. R. Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

2011 (7)

M. V. Romalis and H. B. Dang, “Atomic magnetometers for materials characterization,” Mater. Today 14(6), 258–262 (2011).
[Crossref]

B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, “Dynamical decoupling of a single-electron spin at room temperature,” Phys. Rev. B 83(8), 081201 (2011).
[Crossref]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nat. Lett. 475(7536), 359–363 (2011).
[Crossref]

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

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

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref] [PubMed]

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin, “A single nitrogen-vacancy defect coupled to a nanomechanical oscillator,” Nat. Phys. 7(11), 879–883 (2011).
[Crossref]

2010 (1)

O. Romero-Isart, M. L. Luan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[Crossref]

2009 (3)

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

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity optomechanics using optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107(3), 1005–1010 (2009).
[Crossref]

P. Rabl, P. Cappellaro, M. V. Gurudev Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, “Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,” Phys. Rev. B 79(4), 041302 (2009).
[Crossref]

2008 (3)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-Action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
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J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Tylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin, “Nanoscale magnetic sensing with an individual electronic spin in diamond,” Nat. Lett. 455, 644–647 (2008).
[Crossref]

J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, “High-sensitivity diamond magnetometer with nanoscale resolution,” Nat. Phys. 4, 810–816 (2008).
[Crossref]

2007 (3)

D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
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A. Trabesinger, “Magnetic resonance imaging: Take a close look,” Nat. Phys. 3, 302 (2007).
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B. Polyak, I. Fishbein, M. Chorny, I. Alferiev, D. Williams, B. Yellen, G. Friedman, and R. J. Levy, “High field gradient targeting of magnetic nanoparticle-loaded endothelial cells to the surfaces of steel stents,” Proc. Nat. Acad. Sci. U.S.A. 105(2), 698–703 (2007).
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2005 (1)

I. M. Savukov, S. J. Seltzer, M. V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
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2004 (1)

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nat. Lett. 430, 329–332 (2004).
[Crossref]

2002 (2)

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65(6), 063803 (2002).
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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(14), 148301 (2002).
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2000 (1)

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
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1993 (1)

W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, “Quantum projection noise: population fluctuations in two-level systems,” Phys. Rev. A 47(5), 3554–3570 (1993).
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Acosta, V. M.

K. Fang, V. M. Acosta, C. Santori, Z. Huang, K. M. Itoh, H. Watanabe, S. Shikata, and R. G. Beausoleil, “High-sensitivity magnetometry based on quantum beats in diamond nitrogen-vacancy centers,” Phys. Rev. Lett. 110(13), 130802 (2013).
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Ahn, J.

T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Qi Yin, and T. Li, “Torsional optomechanics of a levitated nonspherical nanoparticle,” Phys. Rev. Lett. 117(12), 123604 (2016).
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T. M. Hoang, J. Ahn, J. Bang, and T. Li, “Electron spin control of optically levitated nanodiamonds in vacuum,” Nat. Commun. 7, 12250 (2016).
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Albrecht, A.

A. Albrecht, A. Retzker, and M. B. Plenio, “Testing quantum gravity by nanodiamond interferometry with nitrogen-vacancy centers,” Phys. Rev. A 90(3), 033834 (2014).
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Alferiev, I.

B. Polyak, I. Fishbein, M. Chorny, I. Alferiev, D. Williams, B. Yellen, G. Friedman, and R. J. Levy, “High field gradient targeting of magnetic nanoparticle-loaded endothelial cells to the surfaces of steel stents,” Proc. Nat. Acad. Sci. U.S.A. 105(2), 698–703 (2007).
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Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nat. Lett. 475(7536), 359–363 (2011).
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Amezcua, M.

D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical quantum control of nitrogen-vacancy center in diamond,” Phys. Rev. Lett. 116(14), 143602 (2016).
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Arcizet, O.

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin, “A single nitrogen-vacancy defect coupled to a nanomechanical oscillator,” Nat. Phys. 7(11), 879–883 (2011).
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Arita, Y.

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

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(14), 148301 (2002).
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Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
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J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nat. Lett. 478(7367), 89–92 (2011).
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O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref] [PubMed]

Atherton, D. P.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using micro-spheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91(5), 051805 (2015).
[Crossref]

Bang, J.

T. M. Hoang, J. Ahn, J. Bang, and T. Li, “Electron spin control of optically levitated nanodiamonds in vacuum,” Nat. Commun. 7, 12250 (2016).
[Crossref] [PubMed]

T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Qi Yin, and T. Li, “Torsional optomechanics of a levitated nonspherical nanoparticle,” Phys. Rev. Lett. 117(12), 123604 (2016).
[Crossref] [PubMed]

Barker, P. F.

C. Wan, M. Scala, S. Bose, A. C. Frangeskou, ATM A. Rahman, G. W. Morley, P. F. Barker, and M. S. Kim, “Tolerence in the Ramsey interference of a trapped nanodiamond,” Phys. Rev. A 93(4), 043852 (2016).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111(18), 180403 (2013).
[Crossref] [PubMed]

Beausoleil, R. G.

K. Fang, V. M. Acosta, C. Santori, Z. Huang, K. M. Itoh, H. Watanabe, S. Shikata, and R. G. Beausoleil, “High-sensitivity magnetometry based on quantum beats in diamond nitrogen-vacancy centers,” Phys. Rev. Lett. 110(13), 130802 (2013).
[Crossref] [PubMed]

Bennett, S. D.

S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, “Phonon-induced spin-spin interactions in diamond nanostructures: Application to spin squeezing,” Phys. Rev. Lett. 110(15), 156402 (2013).
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S. Kolkowitz, A. C. B. Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. G. E. Harris, and M. D. Lukin, “Coherent sensing of a mechanical resonator with a single-spin qubit,” Science 335(6076), 1603–1606 (2012).
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Bergquist, J. C.

W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, “Quantum projection noise: population fluctuations in two-level systems,” Phys. Rev. A 47(5), 3554–3570 (1993).
[Crossref] [PubMed]

Bhattacharya, M.

W. Ge, B. Rodenburg, and M. Bhattacharya, “Feedback-induced bistability of an optically levitated nanoparticle: A Fokker-Planck treatment,” Phys. Rev. A 94(2), 023808 (2016).
[Crossref]

W. Ge and M. Bhattacharya, “Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence,” New J. Phys. 18, 103002 (2016).
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B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, “Quantum model of cooling and force sensing with an optically trapped nanoparticle,” Optica 3(3), 318 (2016).
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Blaser, F.

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref] [PubMed]

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(14), 148301 (2002).
[Crossref] [PubMed]

Bollinger, J. J.

W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, “Quantum projection noise: population fluctuations in two-level systems,” Phys. Rev. A 47(5), 3554–3570 (1993).
[Crossref] [PubMed]

Bose, S.

C. Wan, M. Scala, S. Bose, A. C. Frangeskou, ATM A. Rahman, G. W. Morley, P. F. Barker, and M. S. Kim, “Tolerence in the Ramsey interference of a trapped nanodiamond,” Phys. Rev. A 93(4), 043852 (2016).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bose, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111(18), 180403 (2013).
[Crossref] [PubMed]

Bowen, W. P.

C. Yu, J. Janousek, E. Sheridan, D. L. McAuslan, H. R. Dunlop, P. K. Lam, Y. Zhang, and W. P. Bowen, “Optomechanical magnetometry with a macroscopic resonator,” Phys. Rev. App. 5(4), 044007 (2016).
[Crossref]

S. Forstner, E. Sheridan, J. Knittel, C. L. Humphreys, G. A. Brawley, H. R. Dunlop, and W. P. Bowen, “Ultrasensitive optomechanical magnetometry,” Adv. Mater. 26(36), 6348–6353 (2014).
[Crossref] [PubMed]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swain, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. R. Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Brawley, G. A.

S. Forstner, E. Sheridan, J. Knittel, C. L. Humphreys, G. A. Brawley, H. R. Dunlop, and W. P. Bowen, “Ultrasensitive optomechanical magnetometry,” Adv. Mater. 26(36), 6348–6353 (2014).
[Crossref] [PubMed]

Brennen, G. K.

M. Cirio, G. K. Brennen, and J. Twamley, “Quantum magnetomechanics: Ultrahigh-Q-levitated mechanical oscillators,” Phys. Rev. Lett. 109(14), 147206 (2012).
[Crossref] [PubMed]

Budakian, R.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nat. Lett. 430, 329–332 (2004).
[Crossref]

Budker, D.

J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, “High-sensitivity diamond magnetometer with nanoscale resolution,” Nat. Phys. 4, 810–816 (2008).
[Crossref]

D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

Cappellaro, P.

P. Rabl, P. Cappellaro, M. V. Gurudev Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, “Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,” Phys. Rev. B 79(4), 041302 (2009).
[Crossref]

J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Tylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin, “Nanoscale magnetic sensing with an individual electronic spin in diamond,” Nat. Lett. 455, 644–647 (2008).
[Crossref]

J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, “High-sensitivity diamond magnetometer with nanoscale resolution,” Nat. Phys. 4, 810–816 (2008).
[Crossref]

Chan, J.

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

Chang, D. E.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity optomechanics using optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107(3), 1005–1010 (2009).
[Crossref]

Chassagneux, Y.

T. Delord, L. Nicolas, Y. Chassagneux, and G. Hétet, “Strong coupling between a single NV spin and the torsional mode of diamonds levitating in an ion trap,” arXiv:1702.00774.

Chen, C.

Q. Hou, W. Yang, C. Chen, and Z. Yin, “Generation of macroscopic Schrödinger cat states in diamond mechanical resonator,” Sci. Rep. 6, 37542 (2016).
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Chen, L.

Childress, L.

J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, “High-sensitivity diamond magnetometer with nanoscale resolution,” Nat. Phys. 4, 810–816 (2008).
[Crossref]

Chorny, M.

B. Polyak, I. Fishbein, M. Chorny, I. Alferiev, D. Williams, B. Yellen, G. Friedman, and R. J. Levy, “High field gradient targeting of magnetic nanoparticle-loaded endothelial cells to the surfaces of steel stents,” Proc. Nat. Acad. Sci. U.S.A. 105(2), 698–703 (2007).
[Crossref]

Chui, B. W.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nat. Lett. 430, 329–332 (2004).
[Crossref]

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nat. Lett. 475(7536), 359–363 (2011).
[Crossref]

Cirac, J. I.

O. Romero-Isart, L. Clemente, C. Navau, A. Sanchez, and J. I. Cirac, “Quantum magnetomechanics with levitating superconducting microspheres,” Phys. Rev. Lett. 109(14), 147205 (2012).
[Crossref] [PubMed]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref] [PubMed]

O. Romero-Isart, M. L. Luan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[Crossref]

Cirio, M.

M. Cirio, G. K. Brennen, and J. Twamley, “Quantum magnetomechanics: Ultrahigh-Q-levitated mechanical oscillators,” Phys. Rev. Lett. 109(14), 147206 (2012).
[Crossref] [PubMed]

Clemente, L.

O. Romero-Isart, L. Clemente, C. Navau, A. Sanchez, and J. I. Cirac, “Quantum magnetomechanics with levitating superconducting microspheres,” Phys. Rev. Lett. 109(14), 147205 (2012).
[Crossref] [PubMed]

Cunningham, M.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using micro-spheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91(5), 051805 (2015).
[Crossref]

Dang, H. B.

M. V. Romalis and H. B. Dang, “Atomic magnetometers for materials characterization,” Mater. Today 14(6), 258–262 (2011).
[Crossref]

Dellago, C.

V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, “Direct measurement of photon recoil from a levitated nanoparticle,” Phys. Rev. Lett. 116(24), 243601 (2016).
[Crossref] [PubMed]

Delord, T.

T. Delord, L. Nicolas, L. Schwab, and G. Hétet, “Electron spin resonance from NV centers in diamonds levitating in an ion trap,” New. J. Phys. 19, 033031 (2017).
[Crossref]

T. Delord, L. Nicolas, Y. Chassagneux, and G. Hétet, “Strong coupling between a single NV spin and the torsional mode of diamonds levitating in an ion trap,” arXiv:1702.00774.

Dholakia, K.

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

Dolde, F.

B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, “Dynamical decoupling of a single-electron spin at room temperature,” Phys. Rev. B 83(8), 081201 (2011).
[Crossref]

Donner, T.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nat. Lett. 475(7536), 359–363 (2011).
[Crossref]

Duan, L. M.

Z. Yin, T. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88(3), 033614 (2013).
[Crossref]

Dunlop, H. R.

C. Yu, J. Janousek, E. Sheridan, D. L. McAuslan, H. R. Dunlop, P. K. Lam, Y. Zhang, and W. P. Bowen, “Optomechanical magnetometry with a macroscopic resonator,” Phys. Rev. App. 5(4), 044007 (2016).
[Crossref]

S. Forstner, E. Sheridan, J. Knittel, C. L. Humphreys, G. A. Brawley, H. R. Dunlop, and W. P. Bowen, “Ultrasensitive optomechanical magnetometry,” Adv. Mater. 26(36), 6348–6353 (2014).
[Crossref] [PubMed]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swain, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. R. Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Dutt, M. V. G.

J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Tylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin, “Nanoscale magnetic sensing with an individual electronic spin in diamond,” Nat. Lett. 455, 644–647 (2008).
[Crossref]

Fang, K.

K. Fang, V. M. Acosta, C. Santori, Z. Huang, K. M. Itoh, H. Watanabe, S. Shikata, and R. G. Beausoleil, “High-sensitivity magnetometry based on quantum beats in diamond nitrogen-vacancy centers,” Phys. Rev. Lett. 110(13), 130802 (2013).
[Crossref] [PubMed]

Fedder, H.

B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, “Dynamical decoupling of a single-electron spin at room temperature,” Phys. Rev. B 83(8), 081201 (2011).
[Crossref]

Feng, M.

Fishbein, I.

B. Polyak, I. Fishbein, M. Chorny, I. Alferiev, D. Williams, B. Yellen, G. Friedman, and R. J. Levy, “High field gradient targeting of magnetic nanoparticle-loaded endothelial cells to the surfaces of steel stents,” Proc. Nat. Acad. Sci. U.S.A. 105(2), 698–703 (2007).
[Crossref]

Forstner, S.

S. Forstner, E. Sheridan, J. Knittel, C. L. Humphreys, G. A. Brawley, H. R. Dunlop, and W. P. Bowen, “Ultrasensitive optomechanical magnetometry,” Adv. Mater. 26(36), 6348–6353 (2014).
[Crossref] [PubMed]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swain, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. R. Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Frangeskou, A. C.

C. Wan, M. Scala, S. Bose, A. C. Frangeskou, ATM A. Rahman, G. W. Morley, P. F. Barker, and M. S. Kim, “Tolerence in the Ramsey interference of a trapped nanodiamond,” Phys. Rev. A 93(4), 043852 (2016).
[Crossref]

Friedman, G.

B. Polyak, I. Fishbein, M. Chorny, I. Alferiev, D. Williams, B. Yellen, G. Friedman, and R. J. Levy, “High field gradient targeting of magnetic nanoparticle-loaded endothelial cells to the surfaces of steel stents,” Proc. Nat. Acad. Sci. U.S.A. 105(2), 698–703 (2007).
[Crossref]

Ge, W.

W. Ge and M. Bhattacharya, “Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence,” New J. Phys. 18, 103002 (2016).
[Crossref]

W. Ge, B. Rodenburg, and M. Bhattacharya, “Feedback-induced bistability of an optically levitated nanoparticle: A Fokker-Planck treatment,” Phys. Rev. A 94(2), 023808 (2016).
[Crossref]

Geraci, A. A.

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using micro-spheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91(5), 051805 (2015).
[Crossref]

Z. Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27(26), 1330018 (2013).
[Crossref]

Gieseler, J.

V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, “Direct measurement of photon recoil from a levitated nanoparticle,” Phys. Rev. Lett. 116(24), 243601 (2016).
[Crossref] [PubMed]

Gilligan, J. M.

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W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, “Quantum projection noise: population fluctuations in two-level systems,” Phys. Rev. A 47(5), 3554–3570 (1993).
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Wrachtrup, J.

B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, “Dynamical decoupling of a single-electron spin at room temperature,” Phys. Rev. B 83(8), 081201 (2011).
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Xia, K.

K. Xia and J. Twamley, “Generating spin squeezing states and Greenberger-Horne-Zeilinger entanglement using hybrid phonon-spin ensemble in diamond,” Phys. Rev. B 94(20), 205118 (2016).
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Xu, X.

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90(4), 043848 (2014).
[Crossref]

Yacoby, A.

J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, “High-sensitivity diamond magnetometer with nanoscale resolution,” Nat. Phys. 4, 810–816 (2008).
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J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Tylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin, “Nanoscale magnetic sensing with an individual electronic spin in diamond,” Nat. Lett. 455, 644–647 (2008).
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Q. Hou, W. Yang, C. Chen, and Z. Yin, “Generation of macroscopic Schrödinger cat states in diamond mechanical resonator,” Sci. Rep. 6, 37542 (2016).
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J. Q. Zhang, S. Zhang, J. H. Zou, L. Chen, W. Yang, Y. Li, and M. Feng, “Fast optical cooling of nanomechanical cantilever with the dynamic zeeman effect,” Opt. Express 21(24), 29695–29710 (2013).
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S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, “Phonon-induced spin-spin interactions in diamond nanostructures: Application to spin squeezing,” Phys. Rev. Lett. 110(15), 156402 (2013).
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Yashchuk, V. V.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
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Ye, J.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity optomechanics using optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107(3), 1005–1010 (2009).
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Yellen, B.

B. Polyak, I. Fishbein, M. Chorny, I. Alferiev, D. Williams, B. Yellen, G. Friedman, and R. J. Levy, “High field gradient targeting of magnetic nanoparticle-loaded endothelial cells to the surfaces of steel stents,” Proc. Nat. Acad. Sci. U.S.A. 105(2), 698–703 (2007).
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Yin, Z.

Q. Hou, W. Yang, C. Chen, and Z. Yin, “Generation of macroscopic Schrödinger cat states in diamond mechanical resonator,” Sci. Rep. 6, 37542 (2016).
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Z. Yin, T. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88(3), 033614 (2013).
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Yin, Z. Q.

Z. Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27(26), 1330018 (2013).
[Crossref]

Yin, Z.-q

Y. Ma, T. M. Hoang, M. Gong, T. Li, and Z.-q Yin, “Quantum many-body simulation and torsional matter-wave interferometry with a levitated nanodiamond,” arxiv:1611.05599.

Yin, Z.-Qi

T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Qi Yin, and T. Li, “Torsional optomechanics of a levitated nonspherical nanoparticle,” Phys. Rev. Lett. 117(12), 123604 (2016).
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C. Yu, J. Janousek, E. Sheridan, D. L. McAuslan, H. R. Dunlop, P. K. Lam, Y. Zhang, and W. P. Bowen, “Optomechanical magnetometry with a macroscopic resonator,” Phys. Rev. App. 5(4), 044007 (2016).
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Zhang, J. Q.

Zhang, S.

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Z. Yin, T. Li, X. Zhang, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A 88(3), 033614 (2013).
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Zibrov, A. S.

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Zoller, P.

S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, “Phonon-induced spin-spin interactions in diamond nanostructures: Application to spin squeezing,” Phys. Rev. Lett. 110(15), 156402 (2013).
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Z. Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27(26), 1330018 (2013).
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G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, “Attonewton force detection using micro-spheres in a dual-beam optical trap in high vacuum,” Phys. Rev. A 91(5), 051805 (2015).
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X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90(4), 043848 (2014).
[Crossref]

Phys. Rev. App. (1)

C. Yu, J. Janousek, E. Sheridan, D. L. McAuslan, H. R. Dunlop, P. K. Lam, Y. Zhang, and W. P. Bowen, “Optomechanical magnetometry with a macroscopic resonator,” Phys. Rev. App. 5(4), 044007 (2016).
[Crossref]

Phys. Rev. B (3)

K. Xia and J. Twamley, “Generating spin squeezing states and Greenberger-Horne-Zeilinger entanglement using hybrid phonon-spin ensemble in diamond,” Phys. Rev. B 94(20), 205118 (2016).
[Crossref]

B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, “Dynamical decoupling of a single-electron spin at room temperature,” Phys. Rev. B 83(8), 081201 (2011).
[Crossref]

P. Rabl, P. Cappellaro, M. V. Gurudev Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, “Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,” Phys. Rev. B 79(4), 041302 (2009).
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Phys. Rev. Lett. (13)

S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, “Phonon-induced spin-spin interactions in diamond nanostructures: Application to spin squeezing,” Phys. Rev. Lett. 110(15), 156402 (2013).
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D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical quantum control of nitrogen-vacancy center in diamond,” Phys. Rev. Lett. 116(14), 143602 (2016).
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T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Qi Yin, and T. Li, “Torsional optomechanics of a levitated nonspherical nanoparticle,” Phys. Rev. Lett. 117(12), 123604 (2016).
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Physics (1)

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
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Proc. Nat. Acad. Sci. U.S.A. (1)

B. Polyak, I. Fishbein, M. Chorny, I. Alferiev, D. Williams, B. Yellen, G. Friedman, and R. J. Levy, “High field gradient targeting of magnetic nanoparticle-loaded endothelial cells to the surfaces of steel stents,” Proc. Nat. Acad. Sci. U.S.A. 105(2), 698–703 (2007).
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Proc. Natl. Acad. Sci. U.S.A. (1)

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity optomechanics using optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107(3), 1005–1010 (2009).
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Q. Hou, W. Yang, C. Chen, and Z. Yin, “Generation of macroscopic Schrödinger cat states in diamond mechanical resonator,” Sci. Rep. 6, 37542 (2016).
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S. Kolkowitz, A. C. B. Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. G. E. Harris, and M. D. Lukin, “Coherent sensing of a mechanical resonator with a single-spin qubit,” Science 335(6076), 1603–1606 (2012).
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L. M. Pham, “Magnetic field sensing with nitrogen-vacancy color centers in diamond,” (Doctoral Dissertation, Harvard Uni., 2013).

T. Delord, L. Nicolas, Y. Chassagneux, and G. Hétet, “Strong coupling between a single NV spin and the torsional mode of diamonds levitating in an ion trap,” arXiv:1702.00774.

Y. Ma, T. M. Hoang, M. Gong, T. Li, and Z.-q Yin, “Quantum many-body simulation and torsional matter-wave interferometry with a levitated nanodiamond,” arxiv:1611.05599.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the optically levitated nanodiamond oscillating in dipole trap along z-direction. A green laser of wavelength 532 nm excites the nanodiamond. A MFG along z-direction is applied to engineer spin-mechanical coupling. Microwaves are used to manipulate the ground spin state of nanodiamond.

Fig. 2
Fig. 2

(a) Level diagram for excitation and read-out of NV-spins by optical means. Inset shows the bare energy-level diagram for the triplet ground-state of NV-center in nanodiamond. The microwave fields of Rabi frequency Ω0 drives the transitions |0〉 → |+ 1〉 and |0〉 → | − 1〉 with detuning Δ. (b) The dressed-state description of NV-spin. Phonons of energy ωm drives the transitions |c〉 → |b〉 and |b〉 → |a〉 with effective detunings Δ1 = ωmωbc and Δ2 = ωmωab, respectively.

Fig. 3
Fig. 3

(a) Position power spectral density for z-component of motion versus frequency. The parameters chosen are ωm = 38kHz, R =50 nm, density=2200 kg/m3, Teff = 4 K, χ = 10−7, G = 1/18, Ω0 = 0.8ωm, Γ1 = 0.25ωm. (b) Position power spectral density versus scaled spin-mechanical coupling at ω = ωm. The inset shows the position PSD corresponding to red ellipse for smaller values of g/ωm.

Fig. 4
Fig. 4

MFG sensitivity versus (a) frequency (b) scaled spin-mechanical coupling for t m = 2 π ω m. Other parameters are same as in Fig. 3.

Fig. 5
Fig. 5

(a) Steady state phonon number versus g/ωm and Pressure (mbar). MFG sensitivity versus (b)g/ωm and Pressure (mbar), (c) g/ωm and Temperature (K). In plots (a) and (b) Teff =300 K, γopt =1 kHz, while in (c) pressure is 0.3 mbar. The other parameters are same as in Fig. 3.

Fig. 6
Fig. 6

Schematic of the Ramsey microwave pulse sequence.

Fig. 7
Fig. 7

MFG sensitivity versus scaled spin-mechanical coupling. Parameters chosen are C = 5 %, β=10 kcount s−1, and ωm =38 kHz.

Equations (36)

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ρ ˙ = 1 i [ H m , ρ ] ( A t + A p + D p 2 ) D [ Q z ] ρ ( t ) D q 2 D [ P z ] ρ ( t ) i η f 4 m [ Q z , { P z , ρ } ] + [ ρ ( t ) ] ,
| a = sin ( θ ) | 0 + cos ( θ ) | + ; ω a = ( Δ + Δ 2 + 2 Ω 0 2 ) / 2 , | b = | ; ω b = Δ , | c = cos ( θ ) | 0 sin ( θ ) | + ; ω c = ( Δ Δ 2 + 2 Ω 0 2 ) / 2 ,
ρ ˙ = 1 i [ H m , ρ ] ( A t + A p + D p 2 ) D [ Q z ] ρ ( t ) D q 2 D [ P z ] ρ ( t ) i η f 4 m [ Q z , { P z , ρ } ] + [ ρ ( t ) ] A 2 D [ a m ] ρ A + 2 D [ a m ] ρ i δ 2 [ a m a m , ρ ] ,
α 1 = Re [ sin 2 ( θ ) N ( p 2 ρ c c + p 3 ρ c a ) + cos 2 ( θ ) N p 1 ρ b b ] , α 2 = Re [ cos 2 ( θ ) N ( p 1 ρ a a + p 3 ρ a c ) + sin 2 ( θ ) N p 2 ρ b b ] , α 3 = Im [ sin 2 ( θ ) ( p 2 ( ρ c c ρ b b ) + p 3 ρ c a ) N + cos 2 ( θ ) ( p 1 ( ρ a a ρ b b ) + p 3 ρ a c ) N ] ,
Q ˙ z = ( ω m + δ 2 ) P z ( A A + 2 ) Q z ,
P ˙ z = ( ω m + δ 2 ) Q z [ Γ + ( A A + 2 ) ] Q z + 2 m Γ 0 k B T e f f ξ T ( t ) + 18 η Φ n G 2 Q 4 ξ F ( t ) + [ 2 A + + N ( 2 A + 2 A ) ] ξ S ( t ) ,
q ¨ z + [ Γ + A A + ] q ˙ z + { ( ω m + δ 2 ) 2 + [ Γ + ( A A + 2 ) ] ( A A + 2 ) } q z = ( F T + F F + F S ) m ,
S T = ( 1 + δ 2 ω m ) 2 m Γ 0 k B T e f f ,
S F = ( 1 + δ 2 ω m ) 54 m ω m χ 2 Φ G 2 ( 2 N 2 + 2 N + 1 ) ,
S S = ( 1 + δ 2 ω m ) m ω m [ A + + N ( A + A ) ] .
q ˜ z ( ω ) = χ m ( ω ) [ F ˜ T + F ˜ F + F ˜ S ] ,
χ m ( ω ) = 1 m { [ ( ω m + δ 2 ) 2 + ( A A + 2 ) ( Γ + A A + 2 ) ω 2 ] i ( Γ + A A + ) ω } .
| δ q ˜ z ( ω ) | 2 = | χ m ( ω ) | 2 [ | F ˜ T | 2 + | F ˜ F | 2 + | F ˜ S | 2 ] .
| δ q ˜ z ( ω ) | 2 = | χ m ( ω ) | 2 [ S T + S F + S S ] + z 0 2 χ 2 Φ .
α = 4 Γ 1 ( 1 + cos ( 2 θ ) ) sin 2 ( 2 θ ) 9 cos 2 ( 2 θ ) .
| χ m ( ω m ) | m a x 2 = [ m 2 { ( Γ + A ) 2 [ ( ω m 2 + A 2 ( Γ + A 2 ) ) 1 2 ( Γ + A ) 2 ] } ] 1 .
η B = | δ q ˜ z ( ω ) | 2 | δ q ˜ z ( ω ) | 2 B 0 × t m ,
η B = | δ q ˜ z ( ω ) | 2 μ B g l z 0 | χ m ( ω ) | 2 [ D + m ω m N α ] ,
D = 2 m 2 g α ( Γ + A ) | χ m ( ω ) | 2 ( S T + S F + S S ) [ ω m 2 + ω 2 + A 2 ( Γ + A 2 ) ] .
N ˙ = 2 J N 2 ( J + K ) N + ,
N s s D p + γ o p t + A + 2 J .
η B = μ B g l z 0 | δ q ˜ z ( ω ) | 2 | χ m ( ω ) | 2 ( D 1 + D 2 + D 3 ) + D 4 ( S T + S F + S S ) ,
D 1 = 2 g α 3 ω m 2 m Γ 0 K B T e f f ,
D 2 = 108 m g ω m χ 2 Φ G 2 [ α 3 ω m ( 2 N 2 + 2 N + 1 ) + ( 1 + δ 2 ω m ) α 2 J ( 2 + 1 N ) ] ,
D 3 = m ω m 2 g [ α 3 ω m { A + + N ( A + A ) } + ( 1 + δ 2 ω m ) { α 2 A + + N ( α 2 A + α 1 A ) + 1 2 N α 2 J ( A + A ) } ] ,
D 4 = 8 m 2 g | χ m ( ω ) | 4 { [ ( ω m + δ 2 ) 2 + ( A A + 2 ) ( Γ + A A + 2 ) ω 2 ] × [ α 3 ( ω m + δ 2 ) + ( α 1 + α 2 2 ) ( Γ + A A + ) ] + ( α 1 α 2 ) ( Γ + A A + ) ω 2 } .
| ψ 1 = U π / 2 | ψ ( 0 ) = ( | 0 i | + 1 2 ) | α .
| ψ 2 ( t ) = U t | ψ 1 = ( | 0 | α ( t , 0 ) i | + 1 | α ( t , + 1 ) 2 ) ,
| ψ 2 ( T ) = ( | 0 i exp ( 4 i g 2 T ω m ) | + 1 2 ) | α .
| ψ 3 = 1 2 [ ( 1 exp ( 4 i g 2 T ω m ) ) | 0 i ( 1 + exp ( 4 i g 2 T ω m ) ) | 1 ] | α .
P 0 = 1 2 ( 1 cos ( 4 g 2 T ω m ) ) .
S = β 0 + β 1 2 β 0 β 1 2 cos ( 4 g 2 T ω m ) ,
δ B 0 = δ B 0 p s n 2 + δ B 0 s p n 2 ,
( δ B 0 ) min = β + 1 cos 4 ( 4 g 2 T ω m ) max | S B 0 | .
( δ B 0 ) min = ω m β + 1 cos 4 ( 4 g 2 T ω m ) 8 g μ B g l z 0 T C β ,
η B = ( δ B 0 ) min t m ω m β + 1 cos 4 ( 4 g 2 T ω m ) 8 g μ B g l z 0 T C β ,

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