Abstract

Ultrasensitively weighing nanoparticle masses is at the heart of modern measurement techniques. The traditional electrical mass measurement techniques, which are supported by external circuits, have been well known since the previous decade. In the present article, based on the all-optical technique, we propose a scheme of an optical sensor to weigh the masses of nanoparticles via a doubly clamped suspended carbon nanotube resonator. By measuring the resonance frequency shift of the nanotube in the probe absorption spectrum, we can easily determine the masses of external particles landing onto the surface of nanotube. This mass sensor may lead to a novel ultrasensitive measurement technique in nanoscience.

© 2012 Optical Society of America

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2011 (3)

J. J. Li and K. D. Zhu, “Plasmon-assisted mass sensing in a hybrid nanocrystal coupled to a nanomechanical resonator,” Phys. Rev. B 83, 245421 (2011).
[CrossRef]

C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system” J. Appl. Phys. 110, 083107 (2011).
[CrossRef]

J. J. Li and K. D. Zhu, “All-optical Kerr modulator based on a carbon nanotube resonator,” Phys. Rev. B 83, 115445 (2011).
[CrossRef]

2010 (1)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

2009 (3)

P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, “Anomalous dissipation in single-walled carbon nanotube resonators,” Nano Lett. 9, 3699–3703 (2009).
[CrossRef]

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[CrossRef]

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[CrossRef]

2008 (4)

B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, “Ultrasensitive mass sensing with a nanotube electromechanical resonator,” Nano Lett. 8, 3735–3738 (2008).
[CrossRef]

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nature Nanotechnol. 3, 533–537 (2008).
[CrossRef]

I. Wilson-Rae, “Intrinsic dissipation in nanomechanical resonators due to phonon tunneling,” Phys. Rev. B 77, 245418 (2008).
[CrossRef]

H. Y. Chiu, P. Hung, H. W. Ch. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[CrossRef]

2006 (3)

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, “Zeptogram-scale nanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef]

N. Sinha, J. Z. Ma, and J. T. W. Yeow, “Carbon nanotube based sensors,” J. Nanosci. Nanotechnol. 6, 573–590 (2006).
[CrossRef]

B. Domon and R. Aebersold, “Mass spectrometry and protein analysis,” Science 312, 212–217 (2006).
[CrossRef]

2005 (2)

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical system,” Rev. Sci. Instrum. 76, 061101 (2005).
[CrossRef]

K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58(7), 36–42 (2005).
[CrossRef]

2004 (5)

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

K. L. Ekinci, X. M. H. Huang, and M. L. Roukes, “Ultrasensitive nanoelectromechanical mass detection,” Appl. Phys. Lett. 84, 4469–4471 (2004).
[CrossRef]

V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. A. Arias, and P. L. McEuen, “A tunable carbon nanotube electromechanical oscillator,” Nature 431, 284–287 (2004).
[CrossRef]

C. Y. Li and T. W. Chou, “Mass detection using carbon nanotube-based nanomechanical resonators,” Appl. Phys. Lett. 84, 5246–5248 (2004).
[CrossRef]

K. L. Ekinci, Y. T. Tang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
[CrossRef]

2001 (1)

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

1991 (1)

J. F. Lam, S. R. Forrest, and G. L. Tangonan, “Optical nonlinearities in crystalline organic multiple quantum wells,” Phys. Rev. Lett. 66, 1614–1617 (1991).
[CrossRef]

Aebersold, R.

B. Domon and R. Aebersold, “Mass spectrometry and protein analysis,” Science 312, 212–217 (2006).
[CrossRef]

Aguasca, A.

B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, “Ultrasensitive mass sensing with a nanotube electromechanical resonator,” Nano Lett. 8, 3735–3738 (2008).
[CrossRef]

Arcizet, Ol.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Arias, T. A.

V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. A. Arias, and P. L. McEuen, “A tunable carbon nanotube electromechanical oscillator,” Nature 431, 284–287 (2004).
[CrossRef]

Bachtold, A.

B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, “Ultrasensitive mass sensing with a nanotube electromechanical resonator,” Nano Lett. 8, 3735–3738 (2008).
[CrossRef]

Bockrath, M.

H. Y. Chiu, P. Hung, H. W. Ch. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008), p. 313.

Budakian, R.

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

Callegari, C.

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, “Zeptogram-scale nanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef]

Calleja, M.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[CrossRef]

Chen, B.

C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system” J. Appl. Phys. 110, 083107 (2011).
[CrossRef]

Chiu, H. Y.

H. Y. Chiu, P. Hung, H. W. Ch. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[CrossRef]

Chou, T. W.

C. Y. Li and T. W. Chou, “Mass detection using carbon nanotube-based nanomechanical resonators,” Appl. Phys. Lett. 84, 5246–5248 (2004).
[CrossRef]

Chui, B. W.

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

Cicero, G.

P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, “Anomalous dissipation in single-walled carbon nanotube resonators,” Nano Lett. 9, 3699–3703 (2009).
[CrossRef]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Domon, B.

B. Domon and R. Aebersold, “Mass spectrometry and protein analysis,” Science 312, 212–217 (2006).
[CrossRef]

Ekinci, K. L.

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, “Zeptogram-scale nanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef]

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical system,” Rev. Sci. Instrum. 76, 061101 (2005).
[CrossRef]

K. L. Ekinci, Y. T. Tang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
[CrossRef]

K. L. Ekinci, X. M. H. Huang, and M. L. Roukes, “Ultrasensitive nanoelectromechanical mass detection,” Appl. Phys. Lett. 84, 4469–4471 (2004).
[CrossRef]

Feng, X. L.

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[CrossRef]

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, “Zeptogram-scale nanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef]

Forrest, S. R.

J. F. Lam, S. R. Forrest, and G. L. Tangonan, “Optical nonlinearities in crystalline organic multiple quantum wells,” Phys. Rev. Lett. 66, 1614–1617 (1991).
[CrossRef]

Garcia-Sanchez, D.

B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, “Ultrasensitive mass sensing with a nanotube electromechanical resonator,” Nano Lett. 8, 3735–3738 (2008).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed. (Springer, 2000) pp. 425–433.

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Gil-Santos, E.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[CrossRef]

Giovannetti, V.

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

Graff, K. F.

K. F. Graff, Wave Motion in Elastic Solids (Dover, 1991), pp. 539–564.

Greaney, P. A.

P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, “Anomalous dissipation in single-walled carbon nanotube resonators,” Nano Lett. 9, 3699–3703 (2009).
[CrossRef]

Grossman, J. C.

P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, “Anomalous dissipation in single-walled carbon nanotube resonators,” Nano Lett. 9, 3699–3703 (2009).
[CrossRef]

Hanay, M. S.

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[CrossRef]

Hiebert, W. K.

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[CrossRef]

Huang, X. M. H.

K. L. Ekinci, X. M. H. Huang, and M. L. Roukes, “Ultrasensitive nanoelectromechanical mass detection,” Appl. Phys. Lett. 84, 4469–4471 (2004).
[CrossRef]

Hung, P.

H. Y. Chiu, P. Hung, H. W. Ch. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[CrossRef]

Jana, A.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[CrossRef]

Jensen, K.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nature Nanotechnol. 3, 533–537 (2008).
[CrossRef]

Jiang, C.

C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system” J. Appl. Phys. 110, 083107 (2011).
[CrossRef]

Kim, K.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nature Nanotechnol. 3, 533–537 (2008).
[CrossRef]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Lam, J. F.

J. F. Lam, S. R. Forrest, and G. L. Tangonan, “Optical nonlinearities in crystalline organic multiple quantum wells,” Phys. Rev. Lett. 66, 1614–1617 (1991).
[CrossRef]

Lani, G.

P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, “Anomalous dissipation in single-walled carbon nanotube resonators,” Nano Lett. 9, 3699–3703 (2009).
[CrossRef]

Lassagne, B.

B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, “Ultrasensitive mass sensing with a nanotube electromechanical resonator,” Nano Lett. 8, 3735–3738 (2008).
[CrossRef]

Li, C. Y.

C. Y. Li and T. W. Chou, “Mass detection using carbon nanotube-based nanomechanical resonators,” Appl. Phys. Lett. 84, 5246–5248 (2004).
[CrossRef]

Li, J. J.

J. J. Li and K. D. Zhu, “Plasmon-assisted mass sensing in a hybrid nanocrystal coupled to a nanomechanical resonator,” Phys. Rev. B 83, 245421 (2011).
[CrossRef]

J. J. Li and K. D. Zhu, “All-optical Kerr modulator based on a carbon nanotube resonator,” Phys. Rev. B 83, 115445 (2011).
[CrossRef]

C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system” J. Appl. Phys. 110, 083107 (2011).
[CrossRef]

Ma, J. Z.

N. Sinha, J. Z. Ma, and J. T. W. Yeow, “Carbon nanotube based sensors,” J. Nanosci. Nanotechnol. 6, 573–590 (2006).
[CrossRef]

Mamin, H. J.

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

McEuen, P. L.

V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. A. Arias, and P. L. McEuen, “A tunable carbon nanotube electromechanical oscillator,” Nature 431, 284–287 (2004).
[CrossRef]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994), pp. 245–265.

Naik, A. K.

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[CrossRef]

Postma, H. W. Ch.

H. Y. Chiu, P. Hung, H. W. Ch. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[CrossRef]

Raman, A.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[CrossRef]

Ramos, D.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[CrossRef]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Roukes, M. L.

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[CrossRef]

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, “Zeptogram-scale nanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef]

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical system,” Rev. Sci. Instrum. 76, 061101 (2005).
[CrossRef]

K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58(7), 36–42 (2005).
[CrossRef]

K. L. Ekinci, X. M. H. Huang, and M. L. Roukes, “Ultrasensitive nanoelectromechanical mass detection,” Appl. Phys. Lett. 84, 4469–4471 (2004).
[CrossRef]

K. L. Ekinci, Y. T. Tang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
[CrossRef]

Roundy, D.

V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. A. Arias, and P. L. McEuen, “A tunable carbon nanotube electromechanical oscillator,” Nature 431, 284–287 (2004).
[CrossRef]

Rugar, D.

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

Sazonova, V.

V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. A. Arias, and P. L. McEuen, “A tunable carbon nanotube electromechanical oscillator,” Nature 431, 284–287 (2004).
[CrossRef]

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Schwab, K. C.

K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58(7), 36–42 (2005).
[CrossRef]

Sinha, N.

N. Sinha, J. Z. Ma, and J. T. W. Yeow, “Carbon nanotube based sensors,” J. Nanosci. Nanotechnol. 6, 573–590 (2006).
[CrossRef]

Tamayo, J.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[CrossRef]

Tang, Y. T.

K. L. Ekinci, Y. T. Tang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
[CrossRef]

Tangonan, G. L.

J. F. Lam, S. R. Forrest, and G. L. Tangonan, “Optical nonlinearities in crystalline organic multiple quantum wells,” Phys. Rev. Lett. 66, 1614–1617 (1991).
[CrossRef]

Üstünel, H.

V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. A. Arias, and P. L. McEuen, “A tunable carbon nanotube electromechanical oscillator,” Nature 431, 284–287 (2004).
[CrossRef]

Vitali, D.

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

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994), pp. 245–265.

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, Ol. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Wilson-Rae, I.

I. Wilson-Rae, “Intrinsic dissipation in nanomechanical resonators due to phonon tunneling,” Phys. Rev. B 77, 245418 (2008).
[CrossRef]

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[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram of mass sensor with a doubly clamped suspended CNT in the presence of a strong pump beam and a weak probe beam. The chromium atoms or xenon atoms are deposited onto the surface of the nanotube resonator in a special evaporator. The inset shows the energy levels of CNT resonator.

Fig. 2.
Fig. 2.

(a) Probe absorption spectrum with and without landing the external atoms onto the surface of the CNT resonator. The parameters used are Δ pu = 0 , τ n = 3 μs , η = 0.17 , m n = 1580 zg , ω n = 330 MHz , and Ω 2 = 0.03 ( GHz ) 2 (corresponding pump power for a diffraction-limited spot size is below 1 nW). The black curve shows the initial resonance of the CNT, the green curve after landing Cr atoms Δ m = 100 zg , and the red curve after landing Xe atoms Δ m = 220 zg . The top-right inset curve shows the bandwidth of absorption peak, and the bottom-right plot exhibits the relationship between the frequency shift of the CNT and numbers of deposited atoms. (b) Energy levels of the coupled system corresponding to the three peaks in (a).

Fig. 3.
Fig. 3.

Probe absorption spectrum with (a) three different vibrational lifetimes τ n of the CNT for Γ 2 = 50 MHz and (b) three different dephasing rates Γ 2 of the CNT exciton for τ n = 3 μs . The other parameters are the same as Fig. 2(a).

Equations (16)

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H = H ex + H n + H ex n + H ex p = ω ex S z + ω n b + b + ω n η S z ( b + + b ) μ ( S + E pu e i ω pu t + S E pu * e i ω pu t ) μ ( S + E pr e i ω pr t + S E pr * e i ω pr t ) ,
η = 2 3 / 4 ( 1 + σ ) g σ G 1 / 4 ξ ε R ( E h ) 3 / 4 ( q 0 L ) L π cos 3 θ ,
H = Δ pu S z + ω n b + b + ω n β S z ( b + + b ) ( Ω S + + Ω * S ) μ ( S + E pr e i δ t + S E pr * e i δ t ) ,
d d t S z = Γ 1 ( S z + 1 2 ) + i Ω S + i Ω * S + i μ E pr e i δ t S + i μ E pr * e i δ t S ,
d d t S = ( i Δ pu + Γ 2 ) S i ω n η Q S 2 i Ω S z 2 i μ E pr e i δ t S z + F e ^ ,
d 2 d t 2 Q + 1 τ n d d t Q + ω n 2 Q = 2 ω n 2 η S z + ξ ^ ,
ξ ^ + ( t ) ξ ^ ( t ) = γ n ω n d ω 2 π ω e i ω ( t t ) [ 1 + coth ( ω 2 k B T ) ] .
S 0 = 2 Ω S 0 z ( Δ pu + ω n η Q 0 ) i Γ 2 , Q 0 = 2 η S 0 z ,
S = S 0 + δ S , S z = S 0 z + δ S z , Q = Q 0 + δ Q .
δ S z ˙ = Γ 1 δ S z + i Ω δ ( S ) * i Ω * δ S + i μ E pr e i δ t δ ( S ) * i μ E pr * e i δ t δ S ,
δ S ˙ = ( i Δ p u + Γ 2 ) δ S i ω n η ( δ S Q 0 + S 0 δ Q ) 2 i Ω δ S z 2 i μ E p r e i δ t δ S z ,
δ Q ˙ + 1 τ n δ Q ˙ + ω n 2 δ Q = 2 ω n 2 η δ S z .
χ eff ( 1 ) ( ω pr ) = μ S + E pr = μ 2 Γ 2 χ ( ω pr ) ,
ζ ( ω pr ) = ω n 0 2 ω n 0 2 i δ 0 / τ n 0 δ 0 2 ,
( w 0 + 1 ) [ ( Δ pu 0 η 2 ω n 0 w 0 ) 2 + 1 ] + 2 Ω R 2 w 0 = 0.
Δ m = 2 m n ω n Δ ω n .

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