Abstract

We propose an efficient optical electromagnetically induced transparency (EIT) cooling scheme for a cantilever with a nitrogen-vacancy center attached in a non-uniform magnetic field using dynamical Zeeman effect. In our scheme, the Zeeman effect combined with the quantum interference effect enhances the desired cooling transition and suppresses the undesired heating transitions. As a result, the cantilever can be cooled down to nearly the vibrational ground state under realistic experimental conditions within a short time. This efficient optical EIT cooling scheme can be reduced to the typical EIT cooling scheme under special conditions.

© 2013 Optical Society of America

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    [CrossRef] [PubMed]
  46. T. Ishikawa, K.-M. C. Fu, C. Santori, V. M. Acosta, R. G. Beausoleil, H. Watanabe, S. Shikata, and K. M. Itoh, “Optical and spin coherence properties of nitrogen-vacancy centers placed in a 100 nm thick isotopically purified diamond layer,” Nano. Lett.12, 2083–2087 (2012).
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2013 (3)

J. -Q. Zhang, Y. Li, and M. Feng, “Cooling a charged mechanical resonator with time-dependent bias gate voltages,” J. Phys.: Condens. Matter25, 142201 (2013).
[CrossRef]

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett.110, 153606 (2013).
[CrossRef]

Z.-Q. Yin, T.-Z Li, X. Zhang, and L. -M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A88, 033614 (2013).
[CrossRef]

2012 (7)

T. Ishikawa, K.-M. C. Fu, C. Santori, V. M. Acosta, R. G. Beausoleil, H. Watanabe, S. Shikata, and K. M. Itoh, “Optical and spin coherence properties of nitrogen-vacancy centers placed in a 100 nm thick isotopically purified diamond layer,” Nano. Lett.12, 2083–2087 (2012).
[CrossRef] [PubMed]

F. Reiter and A. S. Sorensen, “Effective operator formalism for open quantum systems,” Phys. Rev. A85, 032111 (2012).
[CrossRef]

N. M. Nusran, M. Ummal Momeen, and M. V. Gurudev Dutt, “High-dynamic-range magnetometry with a single electronic spin in diamond,” Nat. Nanotech.7, 109–113, (2012)
[CrossRef]

A. Mari and J. Eisert, “Very hot thermal light can significantly cool quantum systems,” Phys. Rev. Lett.108, 120602 (2012).
[CrossRef]

Z. J. Deng, Y. Li, and C. W. Wu, “Performance of a cooling method by quadratic coupling at high temperatures,” Phys. Rev. A85, 025804 (2012),
[CrossRef]

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

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A86, 053806 (2012).
[CrossRef]

2011 (9)

H. T. Tan and G. X. Li, “Multicolor quadripartite entanglement from an optomechanical cavity,” Phys. Rev. A84, 024301 (2011).
[CrossRef]

Y. Li, L. A. Wu, Y. D. Wang, and L. P. Yang, “Nondeterministic ultrafast ground-state cooling of a mechanical resonator,” Phys. Rev. B84, 094502 (2011).
[CrossRef]

Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A83, 043804 (2011).
[CrossRef]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittake, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature475, 359–363 (2011).
[CrossRef] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Grblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature478, 89–92 (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,” Nature Phys.7, 879–883 (2011).
[CrossRef]

E. Togan, Y. Chu, A. Imamoglu, and M. D. Lukin, “Laser cooling and real-time measurement of the nuclear spin environment of a solid-state qubit,” Nature478, 497–501 (2011).
[CrossRef] [PubMed]

J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: the group theoretic approach,” New J. Phys.13, 025025 (2011).
[CrossRef]

Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A83, 054305 (2011).
[CrossRef]

2010 (5)

M. D. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature466, 730–734 (2010).
[CrossRef]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature464, 697–703 (2010).
[CrossRef]

K. Stannigel, P. Rabl, A. S. Sorensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett.105, 220501 (2010).
[CrossRef]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit, ” Nature466, 730–734 (2010).
[CrossRef] [PubMed]

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nature Phys.6, 602–608 (2010).
[CrossRef]

2009 (5)

V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, “Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature,” Phys. Rev. Lett.102, 057403 (2009).
[CrossRef] [PubMed]

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics2, 40 (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. B79, 041302 (2009).
[CrossRef]

K. Xia and J. Evers, “Ground state cooling of a nanomechanical resonator in the nonresolved regime via quantum interference,” Phys. Rev. Lett.103, 227203 (2009).
[CrossRef]

J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and robust laser cooling of trapped systems,” Phys. Rev. Lett.104, 043003 (2009).
[CrossRef]

2008 (4)

M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett.101, 200503 (2008).
[CrossRef] [PubMed]

L. Tetard, A. Passian, K. T. Venmar, R. M. Lynch, B. H. Voy, G. Shekhawat, V. P. Dravid, and T. Thundat, “Imaging nanoparticles in cells by nanomechanical holography,” Nat. Nanotechnol.3, 501–505 (2008).
[CrossRef] [PubMed]

Y. Li, Y. D. Wang, F. Xue, and C. Bruder, “Quantum theory of transmission line resonator-assisted cooling of a micromechanical resonator,” Phys. Rev. B78, 134301 (2008).
[CrossRef]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature452, 72–75 (2008).
[CrossRef] [PubMed]

2007 (4)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett.99, 093901 (2007).
[CrossRef] [PubMed]

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

F. Xue, Y. D. Wang, Y. X. Liu, and F. Nori, “Cooling a micro-mechanical beam by coupling it to a transmission line,” Phys. Rev. B76, 205302 (2007).
[CrossRef]

A. Retzker and M. B. Plenio, “Fast cooling of trapped ions using the dynamical Stark shift,” New J. Phys.9, 279 (2007).
[CrossRef]

2006 (1)

L. F. Wei, Y. X. Liu, C. P. Sun, and F. Nori, “Probing tiny nanomechanical resonator: classical or quantum mechanical?,” Phys. Rev. Lett.97, 237201 (2006).
[CrossRef]

2005 (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media, ” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

P. Rabl, V. Steixner, and P. Zoller, “Quantum-limited velocity readout and quantum feedback cooling of a trapped ion via electromagnetically induced transparency,” Phys. Rev. A.72, 043823 (2005).
[CrossRef]

2004 (1)

I. Wilson-Rae, P. Zoller, and A. Imamoglu, “Laser cooling of a nanomechanical resonator mode to its quantum ground state,” Phys. Rev. Lett.92, 075507 (2004).
[CrossRef] [PubMed]

2003 (1)

G. Morigi, “Cooling atomic motion with quantum interference,” Phys. Rev. A67, 033502 (2003).
[CrossRef]

2001 (1)

F. Mintert and C. Wunderlich, “Ion-trap quantum logic using long-wavelength radiation,” Phys. Rev. Lett.87, 257904 (2001).
[CrossRef] [PubMed]

2000 (2)

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett.85, 5547–5550 (2000).
[CrossRef]

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett.85, 4458–4461 (2000).
[CrossRef] [PubMed]

1992 (1)

J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A46, 2668–2681 (1992).
[CrossRef] [PubMed]

Acosta, V. M.

T. Ishikawa, K.-M. C. Fu, C. Santori, V. M. Acosta, R. G. Beausoleil, H. Watanabe, S. Shikata, and K. M. Itoh, “Optical and spin coherence properties of nitrogen-vacancy centers placed in a 100 nm thick isotopically purified diamond layer,” Nano. Lett.12, 2083–2087 (2012).
[CrossRef] [PubMed]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Grblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature478, 89–92 (2011).
[CrossRef] [PubMed]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittake, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature475, 359–363 (2011).
[CrossRef] [PubMed]

Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature464, 697–703 (2010).
[CrossRef]

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,” Nature Phys.7, 879–883 (2011).
[CrossRef]

Aspelmeyer, M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Grblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature478, 89–92 (2011).
[CrossRef] [PubMed]

Balasubramanian, G.

V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, “Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature,” Phys. Rev. Lett.102, 057403 (2009).
[CrossRef] [PubMed]

Beausoleil, R. G.

T. Ishikawa, K.-M. C. Fu, C. Santori, V. M. Acosta, R. G. Beausoleil, H. Watanabe, S. Shikata, and K. M. Itoh, “Optical and spin coherence properties of nitrogen-vacancy centers placed in a 100 nm thick isotopically purified diamond layer,” Nano. Lett.12, 2083–2087 (2012).
[CrossRef] [PubMed]

Beck, J.

V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, “Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature,” Phys. Rev. Lett.102, 057403 (2009).
[CrossRef] [PubMed]

Bialczak, R. C.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature464, 697–703 (2010).
[CrossRef]

Blatt, R.

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett.85, 5547–5550 (2000).
[CrossRef]

J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A46, 2668–2681 (1992).
[CrossRef] [PubMed]

Bowen, W. P.

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

Braginsky, V. B.

V. B. Braginsky and A. B. Manukin, Measurements of Weak Forces in Physics Experiments, D. H. Douglass, ed. (Chicago University, 1977).

Bruder, C.

Y. Li, Y. D. Wang, F. Xue, and C. Bruder, “Quantum theory of transmission line resonator-assisted cooling of a micromechanical resonator,” Phys. Rev. B78, 134301 (2008).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic illustration of the optical scheme of NV center-assisted scheme of the effective optical EIT cooling. (b) Three internal energy levels of the NV center are coupled by two lasers satisfying the two-photon resonance Δ+ = Δ. The cantilever vibration is coupled to the NV center by a strong MFG. γ+ (γ) is the decay from the excited state |A2〉 to the ground state |− 1〉 (|+ 1〉), which includes the electron leaking from and pumped into the nearly closed three-level for |A2〉 and | ± 1〉 (see Appendix A for details).

Fig. 2
Fig. 2

(a) Absorption spectrum versus the frequency in our scheme. It is demonstrated that there is a dark dip for two-photon resonance in the absorption rate, and the quantum interference can be used to suppress the heating transitions |n〉 → |n〉 and |n〉 → |n + 1〉 and enhance the cooling transition |n〉 → |n−1〉. The parameters are taken as ωm = 2π × 1 MHz, Γ = 15ωm, Ω0 = 8ωm, and Δ = 31ωm; (b) The cooling cycle of our scheme. The action of the external light fields (with the Rabi frequencies Ω0) and the Zeeman effect (with the coupling λ) can only create the red sideband transition |d, n〉 → |+, n − 1〉 since the carrier transition between |d, n〉 and |+, n〉 and a blue sideband transition |d, n〉 and |+, n + 1〉 are suppressed when two applied lasers are tuned to two-photon resonance. Then, if the decay is from |+, n − 1〉 to |d, n − 1〉, one phonon has been lost compared with the initial state, whereas if the transition is |+, n − 1〉 → |d, n〉, the cycle will be repeated.

Fig. 3
Fig. 3

The final phonon number log〈nss versus Q-value and environmental temperature T (mk), where we have ωm = 2π × 1 MHz, Ω0 = 8ωm, Δ = 31ωm, Γ = 15ωm, and η = 0.115 [31, 32, 35, 37, 39, 44].

Fig. 4
Fig. 4

(a) The cooling and heating coefficients A± and the cooling rate W versus the ratio mR = Ω0/ωm. Here the blue dotted, red solid, and black dashed lines correspond to the heating coefficient, cooling coefficient, and cooling rate, respectively. (b) The final average phonon number 〈nss versus the ratio mR. The parameters are taken from Refs. [31, 32, 35, 37, 39, 44] as ωm = 2π × 1 MHz, Γ = 15ωm, T = 20 mK and η = 0.115.

Fig. 5
Fig. 5

The cooling W in units of the magnetic coupling strength λ versus the ratio mR = Ω0/ωm. Here the blue red solid and red dashed lines correspond to the analytic cooling rate W = AA+ for Eq. (10) and the numerical cooling rate from the master equation for Eq. (2), respectively. And the frequency of the cantilever ωm/2π in (a), (b), and (c) are 1MHz, 5MHz, and 10MHz. Here, Γ/2π = 15MHz, λ/2π = 0.1MHz.

Fig. 6
Fig. 6

The final average phonon number 〈nss as a function of the variations around the optimal Rabi frequency Ω0 = mRωm with the detuning Δ = ( m R 2 2 ) ω m / 2. The simulation is made by Eq. (9), where ωm = 2π × 1MHz, Γ = 15ωm, T = 20mK, η = 0.115 [31, 32, 35,37,39,44]. The dash line, dot-dash line and solid line correspond to γm = 0 Hz, γm = 10 Hz, and γm = 100 Hz, respectively.

Fig. 7
Fig. 7

The pumping process and the decays of NV center. The transition from state |0〉 and state |Ey〉 is driven by the pumping light Ωp. Here γ±1 and Γdark are the direct decays from the excited state |A2〉 to the ground states |±1〉 and the metastable state |1A1〉, respectively. Γ±1 and Γ0 are the direct decays from the excited state |Ey〉 to the ground states | ± 1〉 and |0〉, respectively, while Γ op ± 1 is the indirect decay from the state |0〉 to the state | ± 1〉. γs is the decay from the state |S〉 to the state |0〉.

Fig. 8
Fig. 8

The average phonon number 〈n〉 as a function of the time T. Assume the initial average phonon number is 〈n〉 = 3 and the NV center is in state | − 1〉. The signs for crossing, the black dash line and red solid line are simulated with Eqs. (18), (19) and (20), respectively. Here ωm = 2π × 1MHz, Ω0 = 6ωm, Δ = 10ωm, Γ = 15ωm, γ±1 = Γ/2, Γdark = Γ/130, Γ0 = Γ, Γ±1 = Γ/150, γs = Γ/33, γ0 = 0.1Γ, T = 20mK, η = 0.115, Δp = 0MHz, and Ωp = Γ [31, 32, 35, 3537, 39, 44].

Equations (30)

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H = ω m b b + ω A | A 2 A 2 | + g e μ B B ( 0 ) ( | + 1 + 1 | | 1 1 | ) + 1 2 Ω 0 ( | A 2 + 1 | e i ω + t + | A 2 1 | e i ω t + h . c . ) + λ ( | + 1 + 1 | | 1 1 | ) ( b + b ) .
H rot = ω m b b Δ | A 2 A 2 | + 1 2 Ω 0 ( | A 2 + 1 | + | A 2 1 | + h . c . ) + λ ( | + 1 + 1 | | 1 1 | ) ( b + b ) ,
H 0 ω m b b Δ | A 2 A 2 | + 2 2 Ω 0 ( | A 2 b | + h . c . ) ,
V = η ( b b ) ( Ω 0 2 | A 2 d | h . c . ) ,
A ± = 2 Γ η 2 Ω 0 2 ω m 2 Γ 2 ω m 2 + 4 ( Ω 0 2 / 2 ± Δ ω m ω m 2 ) 2 ,
d d t P ( n ) = [ A + ( N ( ω m ) + 1 ) γ m ] [ ( n + 1 ) P ( n + 1 ) n P ( n ) ] + [ A + + N ( ω m ) γ m ] [ n P ( n 1 ) ( n + 1 ) P ( n ) ] ,
d d t n = ( W + γ m ) n + A + + N ( ω m ) γ m ,
n ( t ) = n s s + e ( W + γ m ) t [ N ( ω m ) n s s ] ,
n s s = [ A + + N ( ω m ) γ m ] / ( W + γ m ) A + / W + N ( ω m ) γ m / W ,
A + = η 2 2 m R 2 ω m 2 Γ 4 ( m R 2 2 ) 2 ω m 2 + Γ 2 , A = η 2 2 m R 2 ω m 2 Γ ,
n s s = Γ 2 16 [ ( m R 2 2 ) / 2 ω m ] 2 + N ( ω m ) γ m W = ( Γ 4 Δ ) 2 + N ( ω m ) γ m W .
H p = ω e | E y E y | + ω + 1 | + 1 + 1 | + ω 1 | 1 1 | + Ω p ( | E y 0 | + | E y 0 | ) cos ( ω p t ) ,
H p = H e + H g + V + V + , H e = Δ e | E y E y | , H g = ω ± 1 | ± 1 ± 1 | , V = Ω p 2 | 0 E y | , V + = Ω p 2 | E y 0 |
H N H = H e i 2 ( Γ 0 + Γ 1 + Γ 1 ) ( | E y E y | ) .
H eff = 1 2 V [ H N V 1 + ( H N V 1 ) ] V + + H g = Δ e Ω p 2 4 Δ e 2 + ( Γ 0 + Γ 1 + Γ + 1 ) 2 | 0 0 | + ω ± 1 | ± 1 ± 1 | ; L op k = L k H N V 1 V + L k = Γ k | k E y | ; L op 0 = Γ 0 | 0 E y | | E y E y | Δ e i 1 2 ( Γ 0 + Γ 1 + Γ + 1 ) Ω p 2 | E y 0 | = Γ 0 1 Δ e i 1 2 ( Γ 0 + Γ 1 + Γ + 1 ) Ω p 2 | 0 0 | ; L op ± 1 = Γ ± 1 | ± 1 E y | | E y E y | Δ e i 1 2 ( Γ 0 + Γ 1 + Γ + 1 ) Ω p 2 | E y 0 | = Γ ± 1 1 Δ e i 1 2 ( Γ 0 + Γ 1 + Γ + 1 ) Ω p 2 | ± 1 0 | .
Γ op ± 1 = | ± 1 | Γ ± 1 1 Δ e i 1 2 ( Γ 0 + Γ 1 + Γ + 1 ) Ω p 2 | ± 1 0 | | 0 | 2 = Γ ± 1 Ω p 2 4 Δ e 2 + ( Γ 0 + Γ 1 + Γ + 1 ) 2 .
Γ op 0 = | 0 | Γ 0 1 Δ e i 1 2 ( Γ 0 + Γ 1 + Γ + 1 ) Ω p 2 | 0 0 | | 0 | 2 = Γ 0 Ω p 2 4 Δ e 2 + ( Γ 0 + Γ 1 + Γ + 1 ) 2 .
d d t ρ = i [ H rot , ρ ] + γ m 2 [ b ρ b ρ b b b b ρ ] + ± γ ± 2 [ | ± 1 A 2 | ρ | A 2 ± 1 | ρ | A 2 A 2 | | A 2 A 2 | ρ ] ,
d d t ρ = i [ H rot ω 0 | 0 0 | , ρ ] + γ m 2 [ b ρ b ρ b b b b ρ ] + ± γ ± 1 2 [ | ± 1 A 2 | ρ | A 2 ± 1 | ρ | A 2 A 2 | | A 2 A 2 | ρ ] + γ 0 2 [ | 0 A 2 | ρ | A 2 0 | ρ | A 2 A 2 | | A 2 A 2 | ρ ] + ± Γ op ± 1 2 [ | ± 1 0 | ρ 0 | ± 1 | ρ | 0 0 | | 0 0 | ρ ] ,
d d t ρ = i [ H rot ω s | A 1 1 A 1 1 | + Ω p ( | E y 0 | + | 0 E y | ) , ρ ] + γ m 2 [ b ρ b ρ b b b b ρ ] + ± γ ± 1 2 [ | ± 1 A 2 | ρ | A 2 ± 1 | ρ | A 2 A 2 | | A 2 A 2 | ρ ] + Γ dark 2 [ | A 1 1 A 2 | ρ | A 2 A 1 1 | ρ | A 2 A 2 | | A 2 A 2 | ρ ] + γ s 2 [ | 0 A 1 1 | ρ | A 1 1 0 | ρ | A 1 1 A 1 1 | | A 1 1 A 1 1 | ρ ] , + Γ 0 2 [ | 0 E y | ρ | E y 0 | ρ | E y E y | | E y E y | ρ ] , + ± Γ ± 1 2 [ | ± 1 E y | ρ | E y ± 1 | ρ | E y E y | | E y E y | ρ ] ,
V = i η x 0 X ( Ω 0 2 | A 2 d | h . c . ) ,
S ( ω ) = 1 2 M ω m 0 dt e i ω t F ( t ) F ( 0 ) s s ,
F ( t ) = d d X V | X = 0 = i η 2 x 0 ( Ω 0 | A 2 d | h . c . ) = η 2 x 0 Ω 0 σ y A 2 , d .
d ρ b b d t = 2 Ω 0 2 σ y A 2 , b + Γ b ( 1 ρ b b ρ d d ) , d ρ d d d t = Γ d ( 1 ρ b b ρ d d ) , d σ x b d d t = 2 Ω 0 2 σ y A 2 , d , d σ y b d d t = 2 Ω 0 2 σ x A 2 , d , d σ x A 2 , b d t = Γ 2 σ x A 2 , b + Δ σ y A 2 , b , d σ y A 2 , b d t = Γ 2 σ y A 2 , b 2 Ω 0 ( 2 ρ b b + ρ d d 1 ) Δ σ x A 2 , b , d σ x A 2 , d d t = Γ 2 σ x A 2 , d 2 Ω 0 2 σ y b d + Δ σ y A 2 , d , d σ y A 2 , d d t = Γ 2 σ y A 2 , d + 2 Ω 0 2 σ x b d Δ σ x A 2 , d ,
S ( ω ) = η 2 ( Ω 0 2 ) 2 0 dt e i ω t σ y A 2 , d ( t ) σ y A 2 , d ( 0 ) s s .
d ρ b b ( t ) σ y A 2 , d ( 0 ) s s d t = 2 Ω 0 2 σ y A 2 , b ( t ) σ y A 2 , d ( 0 ) s s + Γ b ( σ y A 2 , d s s ρ b b ( t ) σ y A 2 , d ( 0 ) s s ρ d d ( t ) σ y A 2 , d ( 0 ) s s ) , d ρ d d ( t ) σ y A 2 , d ( 0 ) s s d t = Γ d ( σ y A 2 , d s s ρ b b ( t ) σ y A 2 , d ( 0 ) s s ρ d d ( t ) σ y A 2 , d ( 0 ) s s ) , d σ x b , d ( t ) σ y A 2 , d ( 0 ) s s d t = 2 Ω 0 2 σ y A 2 , d ( t ) σ y A 2 , d ( 0 ) s s , d σ y b , d ( t ) σ y A 2 , d ( 0 ) s s d t = 2 Ω 0 2 σ x A 2 , d ( t ) σ y A 2 , d ( 0 ) s s , d σ x A 2 , b ( t ) σ y A 2 , d ( 0 ) s s d t = Γ 2 σ x A 2 , b ( t ) σ y A 2 , d ( 0 ) s s + Δ ( σ y A 2 , b ( t ) σ y A 2 , d ( 0 ) ) s s , d σ y A 2 , b ( t ) σ y A 2 , d ( 0 ) s s d t = Γ 2 σ y A 2 , b ( t ) σ y A 2 , d ( 0 ) s s + 2 Ω 0 ( 2 ρ b b ( t ) σ y A 2 , d ( 0 ) s s ) + ρ d d ( t ) σ y A 2 , d ( 0 ) s s σ y A 2 , d s s ) Δ σ x A 2 , b ( t ) σ y A 2 , d ( 0 ) s s , d σ x A 2 , d ( t ) σ y A 2 , d ( 0 ) s s d t = Γ 2 σ x A 2 , d ( t ) σ y A 2 , d ( 0 ) s s 2 Ω 0 2 σ y b , d ( t ) σ y A 2 , d ( 0 ) s s + Δ σ y A 2 , d ( t ) σ y A 2 , d ( 0 ) s s , d σ y A 2 , d ( t ) σ y A 2 , d ( 0 ) s s d t = Γ 2 σ y A 2 , d ( t ) σ y A 2 , d ( 0 ) s s + 2 Ω 0 2 σ x b , d ( t ) σ y A 2 , d ( 0 ) s s Δ σ x A 2 , d ( t ) σ y A 2 , d ( 0 ) s s .
f ( t ) F ( ν ) = 0 dt e i ν t f ( t ) ,
f ( t ) d t f ( 0 ) i ν F ( ν ) .
0 dt e i ω t σ y A 2 , d ( t ) σ y A 2 , d ( 0 ) s s = 2 i ω i Γ ω + 2 Δ ω + 2 ω 2 Ω 0 2
A ± = 2 Re { S ( ω m ) } = 2 Γ η 2 Ω 0 2 ω m 2 Γ 2 ω m 2 + 4 [ Ω 0 2 2 ± Δ ω m ω m 2 ] 2 .

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